In this tutorial, we're going to talk about "or" probability for non-overlapping events. Specifically you will focus on:
"Or" means that either one or both of these two events happen. And non-overlapping means they can't both happen at the same time. That sort of eliminates one of the choices here. One or both of these two events is happening. But they can't both happen. What this means is exactly one of them happens.
Let's take a look at an example.
Take two mutually exclusive, or not overlapping events, like selecting a six from a deck, and selecting a face card from a deck of cards. These can't both happen at the same time. If you are selecting a six, then you are not selecting a face card and vice versa.
What's the probability of selecting a six or a face card? There are 16 sixes or face cards. Four sixes and 12 face cards, out of 52.
You could call this four out of 52, plus 12 out of 52. Four out of 52 was the probability of a six. 12 out of 52 was the probability of a face card.
It would appear, that for non-overlapping events, you can calculate the "or" probability as the probability of A or B occurring, and they can't both happen, would be the individual probabilities added together.
It doesn't work if you have overlapping events. Let's take a look.
What the probability of a six or diamond? If you counted out all the things that were either sixes or diamonds, you would end up with 16 out of 52 cards. That's not the same as adding the probability of a six to the probability of a diamond. This would give you 17 out of 52.
This formula only works for non-overlapping events.
If two events A and B, are mutually exclusive, meaning they can't happen at the same time, then the probability that either A or B happens is equal to the probability of A plus the probability of B.
The "Special Addition Rule for Non-overlapping Events" is the probability that either of two non-overlapping events occurs is the sum of their individual probabilities.
You also learned about the special addition rule for mutually exclusive, or non-overlapping events. This is a special addition rule because there will be a different addition rule for if the events are overlapping.
Source: This work adapted from Sophia Author Jonathan Osters.
The probability that either of two non-overlapping events occurs is the sum of their individual probabilities.