Basic Statistics: the 'Product Rule'

In the previous post, we looked at one of the major ways in which to combine two probabilities. The 'sum rule' was the rule to apply when the occurrence of one event precludes the occurrence of the other event - for example "what are the odds of rolling either a three or a four with your next throw?"

This time we're looking at the rule to apply when the occurrence of one rule is independent of the occurrence of the other one. The method we should apply in this case is called the 'product rule', since we multiply the probabilities together. The catch phrase to look for here is "the probability of x and y" (as opposed to the 'sum rule''s "probability of either x or y").

For instance, if we want to know the probability that we will roll "a four and a six" with our next two throws, the answer is:

(1 ÷ 6) x (1 ÷ 6) = 0.028 (i.e. a little less than 3%)

Another example: what is the probability of hitting the bull's eye (in archery) three times in a row, if you have a 1 in 50 chance of doing it with each arrow fired? Each arrow therefore has a 0.02 chance of hitting the bull's eye, so the odds of hitting three bull's eyes in a row must be:

0.02 x 0.02 x 0.02 = 0.000008

... which, at 8 times in a million, is an appropriately small number! (And smaller still if we round off to two significant figures!)