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

"Either/Or" Probability for Overlapping Events

Author: Jonathan Osters

Calculate an "either/or" probability for overlapping events.

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Source: Tables, images, and graphs created by the author

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In this tutorial, you're going to learn about or probability for overlapping events. This means that the probability of either A, or B, or both, occurring. So, let's take a look at the roulette wheel. What's the probability of even on a roulette wheel? This is not an or probability. All I'm asking is what's the probability of spinning an even number.

And so we look in the even circle. And there are 18 numbers that are even on the roulette wheel, out of 38 sectors. Zero and double zero don't count as even. Now what about this? What is the probability of black on the roulette wheel? Well it's the same idea. Just look in the black bubble, it's also 18 out of 38. Just by coincidence it's same number.

Now how about this. What's the probability of even or black? Well what we can do is we can count up the numbers that are in either even or black. So just count them up. One, two, three, four, five, six, seven, eight, nine, ten, et cetera, et cetera, et cetera. And we get there are 26 of them out of the 38. So now the question is, why is the probability of E or B not equal to the probability of E plus probability of B? This was 18/38 and 18/38.

So when you add those together, you should get 36/38. But we don't, we get 28/38. The reason for the miscount is because some of the things were counted twice. Some stuff was counted as even and some stuff was counted again as being black. Whereas we only counted it once here. And this was the right thing to do.

So which values were double counted? It was the even black sectors. So what we're going to do, is we're going to say, OK well, we can add these together. Except we'll subtract out the things that got double counted. So that we only count them once. And the double counts were the E and B, even and black. And so this is our formula. The probability of E or B is equal to the probability of E plus probability of B, minus probability of E and B. Let's illustrate that.

The probability of E is these numbers, these 18 numbers out of the 38, plus these. But notice these numbers in the middle got both circled and boxed. So we only want to count them once. But by putting them here and here, we counted them twice. So we're going to remove one of them, one of those markings. We're going to remove one of them. And so we only count everything once.

Now important to note, this formula also works for non-overlapping events. Since mutually exclusive events, A and B, can't happen at the same time, the probability that both A and B occur is zero. It's impossible for A and B to both happen. So when we put that into the formula, we end up with probability of A plus probability of B minus 0. When it simplifies down, you may recognize this. This was the special addition rule for non-overlapping events. And so what we see is that the special addition rule for non-overlapping events is a special case of the general addition rule, which has the three terms.

So let's look at it in a two way table now. Students in the middle school were asked about their dominant hand. And so they were asked about their grade and their dominant hand. So these are the right-handed sixth graders, left-handed sixth graders, et cetera. So what's the probability that a student is either eighth grade or left-handed?

Let's do it by formula. Probability of eight or left-handed is equal to probability of being an eighth grader plus the probability of being a left-handed student minus the probability of both. Why minus the probability of both? Because we counted the left-handed eighth graders in the eighth grade row and in the left-hand column. We double counted those 11 students. So we only want to count them once. So we add those probabilities altogether. They all have a common denominator of 338, so that's good. And we end up with 147/338.

The other way to approach this is with just adding up the cells that are either left-handed or eighth grade. So, sixth grade left-handers, seventh grade left-handers, eighth grade right-handers, eighth grade left-handers, and eight grade ambidextrous students. Add them all up and you get 147, out of the total number of students, 338.

So to recap, the probability of at least one or both events occurring is equal to the sum of the individual probabilities minus their joint probability. The reason we subtract out the joint probability is because it was counted in both of their individual probabilities. So we don't want to count it twice, we only want it counted once. With mutually exclusive events, the joint probability is zero and it simplifies down to the special addition formula. We can see the addition rule in both Venn diagrams and two way tables. Just like we did in the previous examples. Good luck and we'll see you next time.

Terms to Know
Either/Or Probability for Overlapping Events

The probability that either of two events occurs is equal to the sum of the probabilities of the two events, minus the joint probability of the two events happening together.  Also known as the "General Addition Rule".

Formulas to Know
Either/Or Probability for 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 thin space minus space P left parenthesis A space a n d space B right parenthesis