Common Core: High School - Statistics and Probability : Using Probability to Make Decisions

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Example Question #1 : Using Probability To Make Decisions

A researcher observes a road that splits into two paths that lead to two destinations. The researcher believes that the path that curves to the right is the better path but wants to know if a random sample of people feels the same way. The researcher surveyed people walking down the isolated street that split into two directions. He decided to question ten random people in order to determine whether they would travel on the path that curved to the right or to the left. If the respondents' answers are random, then does the probability distribution graph follow the pattern of a normal distribution?

Possible Answers:

No, it is skewed to the left.

Cannot be determined

Yes

No, it is skewed to the right.

Correct answer:

Yes

Explanation:

In order to solve this problem, we need to discuss probabilities and the generation of probability distribution models. 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 probabilities in a general sense, we need to determine how we can create a probability distribution graph for a probability model. We will use the following equation to calculate the probabilities to be used in this graphical display:

Remember that in combinations and permutations a combination is calculated using the following formula:

Now, we can write the following formula.

In this formula variables are defined in the following manner:

Screen shot 2016 04 13 at 6.28.51 pm

Let's investigate this standard through the use of an example. Suppose a researcher rolls a die twelve times and notes whether the die rolls on an even or an odd number. If the die is fair (i.e. every number has an equal probability of being rolled or each roll is random), then would the probability distribution graph follow the pattern of a normal distribution? Lets create a table and solve for each variable. The probability of rolling an even number—a two, four, or a six— is three out of six or fifty percent. Likewise, the probability of failure (i.e. rolling an odd number) is three out of six or fifty percent. Next, we need to list the number of successes for each event—variable . The researcher can roll an even number every time he rolls and he may not even roll an even number in all twelve trial. We need to calculate this probability for every possible number of successful events; therefore, the number of successes ranges from zero to twelve. Last, we know that there are a total of twelve trials. We have solved for the probability of each of these variables for every possible number of successes in the trials.

Screen shot 2016 04 13 at 6.29.39 pm

Once this data is tabulated we can graph the probability of rolling an even number. If we look at the graph then we can see if it follows the bell shape curve of a normal distribution.

P rolling an even

A bell curve is shaped like the following image:

 Normal

We can quickly tell that the graph of the probability distribution does follow the shape of a normal or "bell" curve.

Let's use this information to solve the question. If we look at the question, then we know that the probability of turning right or left in a random situation is fifty percent; therefore, the probability of success (i.e. the right path) or failure (i.e. the left path) is one half or fifty percent. Next we know that there were ten people questioned—or ten trials—and the number of successes per trial ranged from zero to ten. This information has been tabulated and the probability of choosing the left or right has been calculated.

Screen shot 2016 04 13 at 8.16.30 pm

We can now graph the probabilities. 

P right or left

We can see that the graph follows a normal distribution with a characteristic bell shape.

Example Question #1 : Expected Value Of Random Variable: Ccss.Math.Content.Hss Md.A.2

Find the expected value of Set A.

Possible Answers:

25

19

23.5

8

16

Correct answer:

23.5

Explanation:

 

 

 

Example Question #3 : Using Probability To Make Decisions

An archer is shooting in a bullseye competition. His target is similar to the bullseye image provided.

Bullseye

The provided table contains the average probability of the archer hitting a particular score at twenty yards at any given time. What is the expected value of each of the archer's shots if he obtains the following series of scores:

Screen shot 2016 04 20 at 1.23.33 pm

Possible Answers:

Cannot be determined

Correct answer:

Explanation:

In order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the archer problem, we need to use the expected mean formula. We will substitute each of the archer's scores with its respective probability and solve.

It is important to note that we cannot solve this equation using the following formula:

This is because each shot has a different probability. The archer does not consistently score each value because the values occupy different positions on the target; therefore, there is a different probability for each shot not one out of six as the incorrect formula assumes.

Example Question #4 : Using Probability To Make Decisions

An archer is shooting in a bullseye competition. His target is similar to the bullseye image provided.

Bullseye

The provided table contains the average probability of the archer hitting a particular score at twenty yards at any given time. What is the expected value of each of the archer's shots if he obtains the following series of scores:

Possible Answers:

Correct answer:

Explanation:

n order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the archer problem, we need to use the expected mean formula. We will substitute each of the archer's scores with its respective probability and solve.

It is important to note that we cannot solve this equation using the following formula:

This is because each shot has a different probability. The archer does not consistently score each value because the values occupy different positions on the target; therefore, there is a different probability for each shot not one out of six as the incorrect formula assumes.

Example Question #5 : Using Probability To Make Decisions

An archer is shooting in a bullseye competition. His target is similar to the bullseye image provided.

Bullseye

The provided table contains the average probability of the archer hitting a particular score at twenty yards at any given time. What is the expected value of each of the archer's shots if he obtains the following series of scores:

Possible Answers:

Correct answer:

Explanation:

n order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the archer problem, we need to use the expected mean formula. We will substitute each of the archer's scores with its respective probability and solve.

It is important to note that we cannot solve this equation using the following formula:

This is because each shot has a different probability. The archer does not consistently score each value because the values occupy different positions on the target; therefore, there is a different probability for each shot not one out of six as the incorrect formula assumes.

Example Question #6 : Using Probability To Make Decisions

An archer is shooting in a bullseye competition. His target is similar to the bullseye image provided.

Bullseye

The provided table contains the average probability of the archer hitting a particular score at twenty yards at any given time. What is the expected value of each of the archer's shots if he obtains the following series of scores:

Possible Answers:

Correct answer:

Explanation:

In order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the archer problem, we need to use the expected mean formula. We will substitute each of the archer's scores with its respective probability and solve.

It is important to note that we cannot solve this equation using the following formula:

This is because each shot has a different probability. The archer does not consistently score each value because the values occupy different positions on the target; therefore, there is a different probability for each shot not one out of six as the incorrect formula assumes.

Example Question #7 : Using Probability To Make Decisions

A list of individuals and the respective size of their motorcycles in cubic inches is provided. If you were to randomly select any one person from the list, then what size would you expect their motorcycle to be?

Screen shot 2016 04 20 at 1.23.51 pm

Possible Answers:

Cannot be determined

Correct answer:

Explanation:

In order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the motorcycle problem, we need to use the expected mean formula. We will substitute each of the motorcycle owner's engine size with its respective probability and solve.

 Each of the values have the same probability of being chosen—one out of eleven. As a result, we can simplify this equation:

Round to the nearest one's place.

 Note that we can solve this problem in this simplified manner because all of the people had an equal probability of being chosen. If his or her probabilities differed, then we would need to substitute in each person’s respective probability of being chosen.

Example Question #8 : Using Probability To Make Decisions

A list of individuals and the respective size of their motorcycles in cubic inches is provided. If you were to randomly select any one person from the list, then what size would you expect their motorcycle to be?

Possible Answers:

Cannot be determined 

Correct answer:

Explanation:

In order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the motorcycle problem, we need to use the expected mean formula. We will substitute each of the motorcycle owner's engine size with its respective probability and solve.

 Each of the values have the same probability of being chosen—one out of eleven. As a result, we can simplify this equation:

Round to the nearest one's place.

Note that we can solve this problem in this simplified manner because all of the people had an equal probability of being chosen. If his or her probabilities differed, then we would need to substitute in each person’s respective probability of being chosen.

Example Question #9 : Using Probability To Make Decisions

A list of individuals and the respective size of their motorcycles in cubic inches is provided. If you were to randomly select any one person from the list, then what size would you expect their motorcycle to be?

Possible Answers:

Cannot be determined 

Correct answer:

Explanation:

In order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the motorcycle problem, we need to use the expected mean formula. We will substitute each of the motorcycle owner's engine size with its respective probability and solve.

 Each of the values have the same probability of being chosen—one out of eleven. As a result, we can simplify this equation:

Round to the nearest one's place.

Note that we can solve this problem in this simplified manner because all of the people had an equal probability of being chosen. If his or her probabilities differed, then we would need to substitute in each person’s respective probability of being chosen.

Example Question #10 : Using Probability To Make Decisions

An archer is shooting in a bullseye competition. His target is similar to the bullseye image provided.

Bullseye

The provided table contains the average probability of the archer hitting a particular score at twenty yards at any given time. What is the expected value of each of the archer's shots if he obtains the following series of scores:

Possible Answers:

Correct answer:

Explanation:

In order to solve this problem, we need to discuss probabilities and how to calculate expected values. 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. 

According to this logic, we would expect to roll a particular number on a die one out of every six rolls; however, we may roll the same number multiple times or not at all in six rolls. This discrepancy creates a difference between the expected mean and the actual mean. The expected mean is a hypothetical calculation that assumes a very large sample size and no intervening variables (such as differing forces on the roll and changes in the friction of the surface the die is rolled upon between rolls). Under these conditions we can calculate that any number on the die in a "perfect world" should be one out of six. On the other hand the true or actual mean is calculated using "real" date. In these calculations we would roll a die a particular number of times and use it to develop a probability of rolling a particular number. It is important to note that, theoretically, actual means will eventually equal expected means over a large—or near infinite—amount of trials. In other words, over many many rolls we would eventually find that each number on the die has a one in six chance of being rolled.

Let's discuss how expected means are determined. The expected mean is calculated using the following formula:

In this equation the variables are identified as the following:

Let's use this information to solve the problem. In order to solve the archer problem, we need to use the expected mean formula. We will substitute each of the archer's scores with its respective probability and solve.

It is important to note that we cannot solve this equation using the following formula:

This is because each shot has a different probability. The archer does not consistently score each value because the values occupy different positions on the target; therefore, there is a different probability for each shot not one out of six as the incorrect formula assumes.

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