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"Either/Or" Probability for Non-Overlapping Events
Common Core: S.CP.7

"Either/Or" Probability for Non-Overlapping Events

Author: Katherine Williams

Calculate an "OR" probability for non-overlapping events.

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This tutorial covers the special addition rule. First, I'll define what the rule and give the formula, and then walk through two examples. Here, with the special addition rule, it's a rule for finding the probability of event A or event B. So it's an or probability when they are non-overlapping.

So that's really key. The events have to be non-overlapping in order to use the special addition rule. In other cases when they are overlapping, we'd use a different rule that we'll have it covered in a different tutorial.

So the special addition rule says you take the probability of event A and add that to the probability of event B. A shortcut way of writing this is this here-- P parentheses A plus P parentheses B. That means the probability of event A plus the probability of event B-- P of A plus P of B. We'll go through two examples.

So I rewrote the formula for the special addition at the top, because that's the most important part-- P of A plus P of B. Now in our first example, what's the probability of choosing a king or choosing a 5? It's an or problem, so the special addition rule applies.

But are the events non-overlapping? King and a 5, can you be a king and a 5? No, so they're non-overlapping, so then we can use the rule. So I need to know what the probability of event A is, and then add that to the probability of event B.

So the probability of event A, what's the probability of being a king? So there's a deck of cards with 52, and four of them are kings.

So the probability of being a king is 4 out of 52. Then we're going to add the probability of event B. So the choosing a 5 is event B. So what's the probability of choosing a 5?

There's four 5's out of 52 cards total, so 4 out of 52. Now when I'm adding fractions, we add the numerators, and the denominator stay the same. So 4 plus 4 gets me 8. And the denominator is going to stay 52.

Now if the denominators weren't the same, you'd have to convert to equivalent fractions before you could add them. Or you could turn them both into decimals and add the decimals together. In either case, you're going to come up with the same answer-- 8 out of 52, which is equivalent to 4 out of 26, which is equivalent to 2 two out of 13, or as decimal, 8 divided by 52 gets me 0.1538.

So I'm going to round that to 0.154. And then as a percent, 15.4%. You could report as a fraction, or any of the other equivalent factions that I named, as a decimal or whereas as a percent. It's all the same thing.

Example 2, selecting a Monday or selecting a Sunday. So what's the probability of doing one or the other? It's an or probability, so we want to be using the special addition rule. And then Monday and Sunday, are those non-overlapping events? Correct. They are non-overlapping events because you can't be a Monday and be a Sunday-- either one.

So the chance of selecting a Monday then, probability of event A, is one 1 of 7. Monday is one of the seven days total. Or probability means we're adding, so plus the probability of selecting a Sunday. Sunday is one out of seven, so 1 out of 7. So again, we add the numerators, 1 plus 1 is 2, and leave the denominator as a 7. Or, convert 1/7 into a decimal and add those two decimals together.

Now to find out what that is as a decimal, 2 divided by 7, 0.2857. So I'm going to round that, 0.286, which, times by 100-- 28.6%. Now here, our probability is 28.6%.

And so that event is more likely to happen. It's more likely to select a Monday or Sunday than it is to choose a king or a 5. We're adding or probabilities together in both cases because that's what the special addition rule tells us.

This has been your tutorial on the special addition rule.

Terms to Know
Either/Or Probability for Non-Overlapping Events

The probability that either of two non-overlapping events occurs is the sum of their individual probabilities.  Also known as the "Special Addition Rule".

Formulas to Know
Either/Or Probability for Non-Overlapping Events

P left parenthesis A space o r space B right parenthesis space equals space P left parenthesis A right parenthesis space plus thin space P left parenthesis B right parenthesis