Inclusion-Exclusion Principle

The inclusion-exclusion principle, also known as the inclusion-exclusion formula, is a fundamental probability theory principle. It calculates the probability of the union of multiple events.
We have already mentioned in the second chapter of the previous section that if random events do not intersect, then the probability of the union of random events is equal to the sum of the probability of occurrence of each random event separately. But how can we calculate the probability of a union of events when they intersect?

Inclusion-Exclusion formula

Well, we can do it using the following formula:

Let's look at the example. Imagine we have 5 bananas, 3 lemons, 2 yellow radishes, 3 red radishes, and 7 green apples. Calculate the probability of getting a fruit or a yellow item.

As you may recognize, fruit can be a yellow item, so event A (getting a yellow item) and event B (getting a fruit) intersect.

The yellow circle includes all yellow items like radishes, lemons, and bananas, while the blue circle represents all fruits such as bananas, lemons, and apples. Some fruits, like bananas and lemons, can be yellow. The intersection of these circles shows that if we simply add the probabilities, we'll count yellow fruits twice. Hence, it's important to subtract the probability of getting a yellow fruit.

So we can calculate corresponding probability as follows: