### All Common Core: High School - Statistics and Probability Resources

## Example Questions

### Example Question #51 : Using Probability To Make Decisions

An unfair four-sided die is thrown and allowed to land without interruption. The probability and payoff of each side of the die is listed in the provided table.

The game costs to play. Given this information, should you play the game?

**Possible Answers:**

No, because over the long run the cost of playing the game is less than the expected payoff.

Yes, because over the long run the cost of playing the game is less than the expected payoff.

None of these

No, because over the long run the cost of playing the game is greater than the expected payoff.

Yes, because over the long run the cost of playing the game is greater than the expected payoff.

**Correct answer:**

No, because over the long run the cost of playing the game is greater than the expected payoff.

This standard requires you to weigh the possible outcomes of a decision by assigning probabilities to payoff values and calculating the values of expected payoffs. In other words, this standard requires us to use the expected means equation to calculate the expected outcomes of a given game and decide if the game should or should not be played. This means that we must have a firm grasp on calculating probabilities as well as expected means. First, we will discuss 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.

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.

Now, 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:

We can illustrate this using an example problem. 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?

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.

Now, lets use this information to solve the problem. We will use the expected means formula and substitute the variable for the given payoff for the side of a die. Substitute the given information into the formula.

Simplify.

The expected value of the payoff over the long run is greater than the money needed to play: therefore, we can write the following:

The correct choice is "No, because over the long run the cost of playing the game is greater than the expected payoff."

### Example Question #392 : High School: Statistics & Probability

An unfair four-sided die is thrown and allowed to land without interruption. The probability and payoff of each side of the die is listed in the provided table.

The game costs to play. Given this information, should you play the game?

**Possible Answers:**

No, because over the long run the cost of playing the game is less than the expected payoff.

None of these

Yes, because over the long run the cost of playing the game is less than the expected payoff.

Yes, because over the long run the cost of playing the game is greater than the expected payoff.

No, because over the long run the cost of playing the game is greater than the expected payoff.

**Correct answer:**

Yes, because over the long run the cost of playing the game is less than the expected payoff.

This standard requires you to weigh the possible outcomes of a decision by assigning probabilities to payoff values and calculating the values of expected payoffs. In other words, this standard requires us to use the expected means equation to calculate the expected outcomes of a given game and decide if the game should or should not be played. This means that we must have a firm grasp on calculating probabilities as well as expected means. First, we will discuss 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.

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.

Now, 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:

We can illustrate this using an example problem. 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?

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.

Now, lets use this information to solve the problem. We will use the expected means formula and substitute the variable for the given payoff for the side of a die. Substitute the given information into the formula.

Simplify.

The expected value of the payoff over the long run is greater than the money needed to play: therefore, we can write the following:

The correct choice is "Yes, because over the long run the cost of playing the game is less than the expected payoff."

### Example Question #393 : High School: Statistics & Probability

An unfair four-sided die is thrown and allowed to land without interruption. The probability and payoff of each side of the die is listed in the provided table.

The game costs to play. Given this information, should you play the game?

**Possible Answers:**

No, because over the long run the cost of playing the game is less than the expected payoff.

Yes, because over the long run the cost of playing the game is less than the expected payoff.

None of these

Yes, because over the long run the cost of playing the game is greater than the expected payoff.

No, because over the long run the cost of playing the game is greater than the expected payoff.

**Correct answer:**

Yes, because over the long run the cost of playing the game is less than the expected payoff.

This standard requires you to weigh the possible outcomes of a decision by assigning probabilities to payoff values and calculating the values of expected payoffs. In other words, this standard requires us to use the expected means equation to calculate the expected outcomes of a given game and decide if the game should or should not be played. This means that we must have a firm grasp on calculating probabilities as well as expected means. First, we will discuss 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.

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.

Now, 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:

We can illustrate this using an example problem. 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?

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.

Now, lets use this information to solve the problem. We will use the expected means formula and substitute the variable for the given payoff for the side of a die. Substitute the given information into the formula.

Simplify.

The expected value of the payoff over the long run is greater than the money needed to play: therefore, we can write the following:

The correct choice is "Yes, because over the long run the cost of playing the game is less than the expected payoff."