# Common Core: High School - Statistics and Probability : Conditional Probability & the Rules of Probability

## Example Questions

### Example Question #1 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Describe the union and intersection of the subsets of set A.

Explanation:

### Example Question #2 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is royal blue and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage.

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal "heads"? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the "heads" side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are royal blue with V8 engines.

Convert to a percentage.

### Example Question #3 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is royal blue and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage.

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal "heads"? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the "heads" side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are royal blue with V8 engines.

Convert to a percentage.

### Example Question #4 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Candy Apple Red and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Candy Apple Red with V8 engines.

### Example Question #5 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Candy Apple Red and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Candy Apple Red with V8 engines.

### Example Question #1 : Conditional Probability & The Rules Of Probability

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Black Pearl and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Black Pearl with V8 engines.

### Example Question #7 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Candy Apple Red and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Candy Apple Red with V8 engines.

### Example Question #8 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Black Pearl and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Black Pearl with V8 engines.

### Example Question #9 : Describe Events As Subsets Of A Sample Space: Ccss.Math.Content.Hss Cp.A.1

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Black Pearl and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Black Pearl with V8 engines.

### Example Question #1 : Conditional Probability & The Rules Of Probability

Market analysts for an automotive company randomly sampled five hundred new car buyers using a simple survey. This survey recorded the preferences of future consumers. The analysts documented several preferences in the same survey: engine size (i.e. an eight cylinder—V8—or a six cylinder—V6) and paint color. The data are present in the provided table.

What of the following correctly calculates the probability intersection future consumers that will want a car that is Black Pearl and has a V8?

Explanation:

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 can discuss other terms related with probabilities. The complement of an event is the outcomes that do not include the event. In other words, what is the probability of not rolling one? For instance, in the previous example, we wanted to know the probability of rolling a one on a die. The complement of this event would be rolling everything but a one. The event in the complement would be rolling a two, three, four, five, or a six. We can calculate the probability using the following method:

Convert to a percentage

We can also find the complement by subtracting the probability of the event from the maximum probability value of one:

It is important to note that in this case the complement is more likely to occur than the original event.

Next, lets discuss unions and intersections. Regarding probabilities, a union is the likelihood that one event or another will occur. For instance, what is the probability that a person will role a one or a six on a die?

Convert to a percentage.

Last, an intersection is the likelihood that one event will occur with another. What is the probability that we will roll a one and flip a coin to reveal heads? We know that the probability of rolling a one is:

Now, let's determine the probability of flipping a coin to reveal the heads side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Let's look at a table that contains these specific probabilities and the key words that are indicative of each.

Now, let's use this question to solve the problem. The question asks us to find the probability of an intersection: cars that are Black Pearl with V8 engines.