In this tutorial, you're going to learn about “or” probability for overlapping events. Specifically you will focus on:
This means that the probability of either A, or B, or both, occurring.
Look at the roulette wheel.
What's the probability of even on a roulette wheel? This is not an “or” probability. What's the probability of spinning an even number. Look in the even circle.
There are 18 numbers that are even on the roulette wheel, out of 38 sectors. Zero and double zero don't count as even. What is the probability of black on the roulette wheel? It's the same idea. Look in the black bubble, it's also 18 out of 38. Just by coincidence it's same number.
What's the probability of even or black? Count up the numbers that are in either even or black. You get 26 of them out of the 38.
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.
When you add those together, you should get 36/38. But you don't, you get 28/38. The reason for the miscount is because some of the things were counted twice. Some were counted as even and some were counted again as being black. Where it was only counted it once was the right thing to do.
Which values were double counted? It was the even black sectors.
You can add these together, except you'll subtract out the things that got double counted so they are only counted once. The double counts were the E and B, even and black. 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 the squared numbers, the 18 numbers out of the 38, plus those in the middle. But notice these numbers in the middle got both circled and boxed.
You only want to count them once. But by putting them in with the E’s and in with the B’s you counted them twice. Remove one of those markings and only count everything once.
It’s 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.
Put that into the formula and you 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. You’ll find that the special addition rule for non-overlapping events is a special case of the general addition rule, which has the three terms.
Look at it in a two way table now. Students in the middle school were asked about their dominant hand.
There are right-handed sixth graders, left-handed sixth graders, et cetera. What's the probability that a student is either eighth grade or left-handed?
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 you counted the left-handed eighth graders in the eighth grade row and in the left-hand column. We double counted those 11 students. You only want to count them once. Add those probabilities altogether. They all have a common denominator of 338 and you 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.
Sixth grade left-handers, seventh grade left-handers, eighth grade right-handers, eighth grade left-handers, and eighth grade ambidextrous students. This equals 147 out of the total number of students, 338.
Either/or probability for overlapping events calculated by subtracting their joint probability. The reason you subtract out the joint probability is because it was counted in both of their individual probabilities. You don't want to count it twice, you only want it counted once.
You’ll find that the special addition rule for non-overlapping events is a special case of the general addition rule, which has the three terms
With mutually exclusive events, the joint probability is zero and it simplifies down to the special addition formula. You can see the addition rule in both Venn diagrams and two way tables.
Source: This work adapted from Sophia Author Jonathan Osters.
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".