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

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

Now, let's convert this into a percentage:

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

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

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

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

Let's multiply these two probabilities together.

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

Let's compare the two probabilities.

The probabilities are equal, therefore,

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

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

Let's multiply these two probabilities together.

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

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

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

The probabilities are not equal.

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

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

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

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

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

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

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

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 less 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."