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 all outcomes gave you the bottom number of a probability. When an event has two descriptions, counting the top number needs more care. The prize table has 40 items, all equally likely to be handed out. Sixteen are toys, ten are red, and four are red toys included in both counts. Somebody asks for the chance of getting a toy or something red. In probability, 'or' means at least one of the two, including both. Adding the counts is an obvious first step.
Adding gives 16 + 10 = 26, but this counts the four red toys twice: once as toys and once as red prizes. Subtract one copy: 16 + 10 - 4 = 22 prizes are a toy or red or both. As a probability, 22 / 40 = 0.55. The rule is P(A or B) = P(A) + P(B) - P(A and B). The difference between the plain sum and the correct probability is the overlap. So P(A and B) = P(A) + P(B) - P(A or B).
Sometimes there is nothing to subtract. Each of the 40 prizes has exactly one size: 18 small, 14 medium and 8 large. No prize is both small and large, so those events are mutually exclusive with no overlap. The rule's last term is zero, so add directly: P(small or large) = 0.45 + 0.2 = 0.65.
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. Use this total to find a missing probability: large is 1 - 0.45 - 0.35 = 0.2. The same idea gives the chance of an event not happening: 1 minus the chance that it does.
A complete set of outcomes that cannot overlap always adds to 1.
Check overlap: can one outcome belong to both events? Independence asks whether learning one happened changes the other's chance. Its test is P(A and B) = P(A) × P(B). Mutually exclusive events with positive probabilities are dependent; if either has probability 0, they are independent. Cards 1 to 12: above 6 and below 10 cover all cards but share 7, 8 and 9. Missing a real overlap makes an 'or' answer too big; subtracting an absent overlap makes it too small.
P(A or B) = P(A) + P(B) - P(A and B). Exclusive events share nothing, so the last term is zero. Outcomes covering everything without overlap add to 1. Independent is a different idea.
Questions x1 and x2 adapt UBC STAT302 homework problems HW04-01a and HW04-01b from the WeBWorK Open Problem Library; x3 and x4 adapt its UBC STAT302 problem HW03-01. Readings adapt Khan Academy's 'Probability and combinatorics: FAQ'. The rewritten readings were written for this dataset. 'Adding with and without an overlap' and questions x5 to x10 were written for os-taxonomy. Source-version license terms are unknown.
You saw every chance on a spinner add up 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.
34 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.