Conditional Probability & the Rules of Probability - AP Statistics

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

No explanation available

In order to solve problems associated with this standard, we need to understand two primary components: probabilities and the property of independent events. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we can calculate a probability, we can review the property of independent events. The probability of independent events states that two events (A and B) are independent if and only if the probability of event A times the probability of event B equals the probability of the intersection of both events A and B occurring. It can be written as the following expression:

Let's return to the dice example. Is the probability of rolling a one on one die independent to rolling a two on a separate die on the same roll? First, let's calculate the probability of rolling a one.

Now, let's find the probability of rolling a two on a separate die in the same roll.

Let's multiply these two probabilities together.

Now, let's find the probability of rolling both a one and a two in the same roll on two separate dice.

Let's compare the two probabilities.

The probabilities are equal, therefore,

Let's look at another example. A person has a bag full of ten marbles: six white and four black marbles. He wants to know if the probability of pulling out a white marble is independent of pulling out a black marble. Let's start by calculating the probability of pulling out a black marble.

Now let's calculate the probability of pulling out a white marble.

Let's multiply these two probabilities together.

Now let's observe these events occurring at simultaneously. Let's say that a black marble is pulled out of the bag first. The probability of the black marble is as follows:

It is the same as before; however, the probability of pulling out a white marble has changed because one marble has been removed from the bag.

The probability of both occurring can be written as the following:

The probabilities are not equal.

This is because these two events are dependent on one another. When a marble is pulled on the first grab (whether it is black or white) and not replaced, there is a different total number of marbles to be pulled on the second grab. As a result, the probability of pulling a marble of a specific color changes from the initial event to the second event. This scenario could be made independent by either using two separate bags with the same marble composition or by replacing the marbles after they are pulled before pulling another one on a second try. This information has ilustrated the property of independence and how to calculate if events are independent from one another.

Now, we can use this information to solve the problem. The question asks us to determine if the probability of buying a truck is independent of it having manual transmission. In this case, we need to investigate whether or not the following expression is true:

First, let's calculate the probability associated with finding a truck in the lot.

Now let's calculate the probability of a vehicle having a manual transmission.

Now, we need to calculate the probability of both of these events occurring simultaneously.

Now, let's find out if the following expression is true:

Using this information, we know that the correct answer is the following:

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

No explanation available

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now, we will start by constructing a table. The question presents data associated with two different types of variables: grade level and class. There are four grade levels (i.e. freshman, sophomores, juniors, and seniors) and four elective science classes (i.e. astronomy, ecology, physics, and chemistry). We need to make a table that has four rows and four columns. Next, we can input the data contained in the table. Last we will add up each row and column in order to gain totals for each variable. If you have done this properly, then you should have created a table similar to the following:

After, the two-way table has been constructed you can then use it to solve the question: what is the probability that a senior will take astronomy?

First, we need to create the following ratio: number seniors planning to take the astrology class to the total number of seniors.

Substitute values using the table and solve.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events, we can simply add the probabilities, which can be illustrated by the following figure. In this figure, each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and, at times, intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.

In order to solve this problem, we need to discuss probabilities and more specifically probabilities of disjoint and non-disjoint events. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate disjoint and non-disjoint probabilities. We will begin by investigating how we calculate the probability of disjoint or mutually exclusive events. If events are referred to as disjoint or mutually exclusive, then they are independent of one another. In order to calculate the probability of two disjoint events, A and B, we need to calculate the probability of event A occurring and add it to the probability that event B will occur. This is formally written using the following equation:

In the case of disjointed events we can simply add the probabilities which can be illustrated by the following figure. In this figure each event is independent of the other.

Let's look at an example to illustrate this concept. Suppose someone wants to calculate rolling a five or a six on a die. In this problem, the word "or" indicates that we need to compute the probability of both; however, we need to know if the events are disjointed or non-disjointed. Disjointed events are independent of one another or mutually exclusive such as the dice rolls in this example. Let's calculate the probabilities of rolling any particular number on a die.

We can see that the probability of rolling any value is one out of six. Now, let's use the formula to solve for the probability of rolling a five or a six.

Next, we need to discuss how to calculate the probabilities of non-disjoint or non-mutually exclusive events. If events are described in this way, then they can intersect at some point. For example, physical traits are not exclusive of each other. A person can have brown eyes, brown hair, or they may have brown eyes and brown hair at the same time. In other words, just because someone has brown eyes does not mean that they cannot have brown hair (i.e. it could be a variety of colors). Likewise, brown hair does not limit a person's eye color to a shade other than brown; thus, the traits may intersect. This example can be illustrated in the following figure. In this figure, events A and B are not exclusive of one another and at times intersect.

When events, A and B, are non-disjoint we can calculate their probability by adding the probability that event A will occur to the probability that event B will occur and then subtract the intersection of both events A and B. This is formally written using the following formula:

Now let's use this information to solve an example problem. Given the following information what is the probability of a truck being royal blue or having a V8 engine?

First, we know that the key word "or" indicates that we are dealing with disjoint or non-disjoint probabilities; however, we need to determine whether the events of our scenario are mutually exclusive or not. We know that they are non-disjoint because a truck can be both royal blue and have a V8 engine, which represents an intersection of the two events. Let's begin by calculating the probability that a truck is royal blue:

Now, we can calculate the probability that a truck will have a V8 engine.

At this point, an example of a common mistake can be illustrated. If we were to treat these events as mutually exclusive then we would follow the formula that simply adds together the probabilities of the two events. If we add together these probabilities, then we would obtain the following value:

This answer would obviously be incorrect because a probability cannot be greater than one; therefore, we need to subtract the probability of the intersection of the two events. Now, we need to calculate the value of this intersection.

Now, we can create an equation to calculate the probability of the non-mutually exclusive events:

Substitute in values and solve.

Let's use this information to solve the question. We need to find the probability that a car will have a V8 or a manual transmission. First, we need to determine whether or not the events are mutually exclusive. We know that the events are non-disjoint because they intersect (i.e. a car may have both a manual and a V8 engine). Next, we need to create a formula to solve for the probability.