Use P(A or B) = P(A) + P(B) - P(A and B), and see why the subtraction disappears when two events cannot happen together.
Counting outcomes gave you the bottom of a probability. Now the top gets harder, because people ask two things at once. The prize table has 40 items on it, all equally likely to be handed out. Sixteen of them are toys. Ten of them are red. Four are red toys, so they are in both counts. Somebody asks the chance of getting a toy or something red. In probability that 'or' means at least one of the two, which takes in both, and the obvious move is to add.
Adding gives 16 + 10 = 26, and that is wrong. The four red toys were counted once as toys and once as red, so 26 counts them twice. Take one copy back: 16 + 10 - 4 = 22 prizes are a toy or red or both. As probabilities, 22 / 40 = 0.55. The rule behind it is P(A or B) = P(A) + P(B) - P(A and B), and the gap between the plain sum and the truth is the overlap itself.
Sometimes there is nothing to give back. Each of the 40 prizes has exactly one size: 18 small, 14 medium and 8 large. Small and large cannot share a prize, so they are mutually exclusive with no overlap. The rule loses its last term: P(small or large) = 0.45 + 0.2 = 0.65. Adding straight is right here.
Every prize has one size and no other size exists, so the sizes cover the table with no overlaps. Their probabilities total 1: 0.45 + 0.35 + 0.2 = 1. That finds a missing one: large is 1 - 0.45 - 0.35 = 0.2. The same idea gives an event not happening: 1 minus the chance that it does.
So ask first: could both happen to one prize? A toy can also be red, so those overlap. A prize cannot be small and large, so those do not. Independent is different: one happening does not change the chance of the other. A coin toss and a prize draw are independent and can happen together. When both have some chance, exclusive events cannot be independent, since one rules out the other. Covering every outcome is separate from sharing none: above 5 and below 9 cover cards yet share 6, 7 and 8. Adding without subtracting counts the overlap twice. Subtracting when there is none throws away prizes counted only once.
P(A or B) = P(A) + P(B) - P(A and B), where the subtraction undoes the double count of anything sitting in both. Two events that cannot both happen are mutually exclusive, there is nothing in both, and the last term is zero. Independent is different: one happening does not change the chance of the other. When both have some chance, exclusive events cannot be independent, since one rules out the other. A complete set of outcomes that cannot overlap has probabilities adding to 1, which finds a missing one and gives the chance of an event not happening. Always ask first whether both could happen at once.
You saw four spinner colors with chance 0.25 each add to 1. The prize table is the next job. Forty prizes, all equally likely to be handed out: 16 are toys, 10 are red, and 4 of them are red toys. A child asks what the chance is of getting a toy or something red.
Sam adds 16 and 10 and writes 26 out of 40. Why is that too many?
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.