Multiplication is a surprisingly powerful tool in statistics. It enables us to solve a vast amount of problems with relative ease. One thing to remember though is that the multiplication rule, to which I’ll get in a bit, only works for independent events. So let’s talk about those first.
When we roll a dice, there’s a certain probability that the number six will show. This probability does not depend on what number we rolled before. The events “rolling a three” and “rolling a six” are independent in the sense, that the occurrence of the one event does not affect the probability for the other.
Let’s look at a card deck. We draw a card and note it. Afterward, we put it back in the deck and mix the cards. Then we draw another one. Does the event “draw an ace” in the first try affect the event “draw a king” in the second try? It does not, because we put the ace back in the deck and mixed the cards. We basically reset our experiment. In such a case, the events “draw an ace” and “draw a king” are independent.
But what if we don’t put the first card back in the deck? Well, when we take the ace out of the deck, the chance of drawing a king will increase from 4 / 52 (4 kings out of 52 cards) to 4 / 51 (4 kings out of 51 cards). If we don’t do the reset, the events “draw an ace” and “draw a king” are in fact dependent. The occurrence of one changes the probability for the other.
With this in mind, we can turn to our powerful tool called multiplication rule. We start with two independent events, A and B. The probabilities for their occurrence are respectively p(A) and p(B). The multiplication rule states that the probability of both events occurring is simply the product of the probabilities p(A) and p(B). In mathematical terms:
p(A and B) = p(A) · p(B).
A quick look at the dice will make this clear. Let’s take both A and B to be the event “rolling a six”. Obviously they are independent, rolling a six on one try will not change the probability of rolling a six in the following try. So we are allowed to use the multiplication rule here. The probability of rolling a six is 1/6, so p(A) = p(B) = 1/6. Using the multiplication rule, we can calculate the chance of rolling two six in a row: p(A and B) = 1/6 · 1/6 = 1/36. Note that if we took A to be “rolling a six” and B to be “rolling a three”, we would arrive at the same result. The chance of rolling two six in a row is the same as rolling a six and then a three.
Can we also use this on the deck of cards, even if we don’t reset the experiment? Indeed we can. But we have to take into account that the probabilities change as we go along. In more abstract terms, instead of looking at the general events “draw an ace” and “draw a king”, we need to look at the events A = “draw an ace in the first try” and B = “draw a king with one ace missing”. With the order of the events clearly set, there’s no chance of them interfering. The occurrence of both events, first drawing an ace and then drawing a king with the ace missing, has the probability: p(A and B) = p(A) · p(B) = 4/52 · 4/51 = 16/2652 or 1 in about 165 or 0.6 %.