# Common Core: High School - Statistics and Probability : Apply and Interpret the Addition Rule: CCSS.Math.Content.HSS-CP.B.7

## Example Questions

### Example Question #61 : Conditional Probability & The Rules Of Probability

What is the probability of choosing a king OR a spade from a regular deck of cards?

Explanation:

### Example Question #61 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Cannot be determined

Explanation:

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.

### Example Question #71 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Cannot be determined

Explanation:

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.

### Example Question #72 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.

### Example Question #73 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.

### Example Question #74 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.

### Example Question #75 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.

### Example Question #76 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.

### Example Question #77 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.

### Example Question #78 : Conditional Probability & The Rules Of Probability

Given the following data, what is the probability that a car will have a V8 or a manual transmission?

Explanation:

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.