## Additional Rule For Probabilities Details

In mathematics, probability refers to methods of quantifying the chances for an event to happen or not happen based on empirical analysis instead of guesses or hunches. When you try to calculate the odds of something happening, you need to know whether the possible events are mutually exclusive. What this means is that if one event happens, any other possible event can’t occur at the same time.

One example is if you throw a die with six sides. The odd for a two or a four to come out on top is 1/6 for each event. However, you can’t get a two and a four simultaneously, so these events are mutually exclusive. Mutual exclusivity describes events that cannot coincide. If there are mutually exclusive events, there are also non-mutually exclusive events.

Non-mutually exclusive events transpire if there’s at least one possibility that the two possible events overlap. For instance, let’s say that you draw a card from a standard deck of 52 playing cards. You want to calculate the probability of a queen or a heart coming out from all the possible events. Regardless of the result, you realize that there’s a possibility that the queen of hearts to appear, so both events are non-mutually exclusive.

## Example of Additional Rule For Probabilities

Some sources may tell you that there are two formulas useful to calculate probabilities involving two different events. Which one you use depends on whether the related events are mutually exclusive or not. Even though this is true, in reality, you simply need to understand the following formula regardless of the events’ mutual exclusiveness:

• P(A or B) = P(A) + P(B) - P(A and B)

Among the two formulas divided according to the events’ mutual exclusiveness, you can use the one above to calculate the probability of two events if the circumstances are not mutually exclusive. For example, let’s use the one where we were drawing a card from a standard deck. If we are trying to compute the probability of getting a queen or a heart, we can use the formula above. The logic of the formula is that we add the probability of either one of two events happening and subtract it with the possibility of both happening so that we don’t count one event (the queen of hearts appearing) twice.

• P(Queen or Heart) = P(Queen) + P(Heart) - P(Queen and Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13

Thus, the probability of a queen or a heart to come out—including both at the same time—is 4/13. Unsurprisingly, we can use the same formula above for mutually exclusive events (for example, the probability of getting a two or a four from a die with six sides). One thing to remember is that since there’s no possibility that the two events happen simultaneously, the value for P(A and B) will always be zero. In that case, we can trim the formula to simplify things.

• P(2 or 4) = P(2) + P(4) - 0 = 1/6 + 1/6 = 2/6 = 1/3

## Types of Additional Rule For Probabilities

Taking all of the things we’ve learned so far, we can conclude that two derived rules come from the addition rule for probabilities.

• For non-mutually exclusive events: P(A or B) = P(A) + P(B) - P(A and B)
• For mutually exclusive events: P(A or B) = P(A) + P(B)

You may also want to know if there’s a formula for computing the probability for precisely one of two events occurring for non-mutually exclusive events. For instance, let’s consider the previous example of getting a queen or a heart from a standard deck with 52 playing cards. Getting precisely one of two events means that you want to get a queen or a heart, but not both at the same time (queen of hearts). For this, there’s one extra rule:

• P(Exactly for one A or one B) = P(A or B) - P(A and B)

Let’s elaborate on how the formula works for better understanding. First, calculate the probability using the regular formula for non-mutually exclusive events: P(A or B) = P(A) + P(B) - P(A and B). After acquiring the value of P(A or B), subtract it with another P(A and B) to eliminate the possibility of two events occurring concurrently. Let’s apply the formula for getting exactly a queen or a heart, but not both at the same time.

• P(exactly for a Queen or a Heart) = P(Queen or Heart) - P(Queen and Heart) = 16/52 - 1/52 = 15/52