Non-mutually exclusive union

We know how to find the probability of either (at least one) of two mutually exclusive events happening (see here). The formula is P(A)+P(B).

How about if the events are not mutually exclusive?

Formula

The probability of either event A or event B occuring (A union B) is:

P(A\cup B)=P(A)+P(B)-P(A\cap B)

Why subtract the intersection?

The reason we subtract the intersection A\cap B is because we double count it. Anything in both A and B will show up in the probability of both A and B, so we need to subtract one ‘lot’ of that intersection to remove the duplicates.

An example of why we need to

For an example, let’s say we have a fruit bowl, and we have a:

• 70% chance of picking a yellowey-orange fruit
• 60% chance of picking a round fruit

If we just added their probabilities like we do with mutually exclusive events, we’d end up with a probability of 130%… which is impossible.

Finding the probability from a Venn diagram

If we have a Venn diagram showing the frequency of A, the frequency of B and the frequency of A and B (the intersection), we can just add the 3 frequencies together (then divide by the total frequency of everything, to get our probability).

That’s because the frequency shown in the A circle isn’t the frequency of the A, but the frequency of only A and nothing else, so A and not B. The same is true for the B circle. So we’re not double counting anything.

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