Multiply probabilities for events that do not affect each other, and test whether two events really are independent before doing so.
Last stop had one prize and two descriptions, and the word was or. Now there are two different stalls and the word is and. The coconut shy is won by 0.2 of the people who play it. The tin-can alley is won by 0.25 of the people who play that. Somebody buys a go at each and wants the chance of walking away with two prizes. Adding would give 0.45, which is more than either game on its own, and winning twice is clearly harder than winning once.
Follow a hundred players through both stalls. Twenty of them win the coconut shy, because 0.2 of a hundred is twenty. Those twenty now play the tins, where a quarter of players win, so five of the twenty win again. Five out of the original hundred is 0.05. That is what 0.2 × 0.25 gives, and it is what multiplying probabilities means: taking a share of a share, rather than putting two shares side by side. It runs backwards too: given independence, divide P(A and B) by P(A) to recover the other one.
Multiplying is only allowed when the second chance is unchanged by the first. Winning at the coconut shy does nothing to the tin cans: different game, different equipment, and the stallholder has never heard of you. Two events like that are called independent, and the test is exactly that question. Does knowing how the first turned out change the probability of the second? If the answer is no, multiply. With three numbers in front of you, work out P(A) × P(B) and see whether it lands on the P(A and B) you were given.
Here is a pair that fails the test. A drum holds 20 tickets and 5 of them win. The first draw wins with probability 5 / 20 = 0.25. If that ticket wins and is not put back, the drum now holds 19 tickets with 4 winners, so the second draw wins with probability 4 / 19, which is about 0.21. The first result changed the second, so the draws are not independent and 0.25 × 0.25 would be wrong. Put the ticket back and the drum is the one you started with, so those two draws would be independent.
That also settles why and multiplies while or adds. Or widens the target, once any overlap is taken off: more outcomes count, so the answer grows past either event on its own. And narrows it: an outcome has to clear both hurdles, so fewer qualify and the answer drops below either one. Two coin flips make the plainest case. A coin has no memory, so the flips are independent and the two halves multiply. Nothing stops at two: a third independent event multiplies in the same way.
For two independent events, P(A and B) = P(A) × P(B). Independent means knowing how the first turned out leaves the second's probability alone, and that question has to be asked before multiplying. Drawing without putting things back fails it, because taking one out changes what is left. And multiplies because an outcome has to clear both hurdles, so the answer falls below either event on its own, while or adds, once any overlap is taken off, because it widens the target.
The prize table taught you what to do with the word or. Two stalls along, the word changes. The coconut shy is won by 0.2 of the people who play it and the tin-can alley by 0.25. Somebody buys a go at each and asks the chance of taking home two prizes. Leo adds and gets 0.45.
Without doing any arithmetic, why can 0.45 not be right?
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.