SERIES WITH INDEPENDENCE

In other words, you cannot miss. There are two good reasons for thinking that this argument is fishy. First, probability can never exceed 100 percent. Second, there must be some chance, however small, that we could toss a coin eight times and not have it come up heads.

The best way to look at this question is to restate it so that the first two rules can be used. Instead of asking what the probability is that heads will come up at least once, we can ask what the probability is that heads will not come up at least once. To say that heads will not come up even once is equivalent to say- ing that tails will come up eight times in a row. By Rule 2 we know how to compute that probability: It is 1/2 multiplied by itself eight times, and that, as we saw, is 1/256. Finally, by Rule 1 we know that the probability that this will not happen (that heads will come up at least once) is 1 – (1/256). In other words, the probability of tossing heads at least once in eight tosses is 255/256. That comes close to a certainty, but it is not quite a certainty.

We can generalize these results as follows:

The probability that an event will occur at least once in a series of independent trials is 1 minus the probability that it will not occur in that number of trials. Symbol- ically (where n is the number of independent trials):

Pr(h at least once in n trials) = 1 – Pr(not h)n

Strictly speaking, Rule 4 is unnecessary, since it can be derived from Rules 1 and 2, but it is important to know because it blocks a common misunder- standing about probabilities: People often think that something is a sure thing when it is not.