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 truck in the lot.

Now let's calculate the probability of a vehicle having a manual 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: