The Language of Argument

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C HaP Te r 1 1 ■ C h a n c e s

heads at least once, and thus after eight tosses there should be a 400 per-

cent chance. 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 equiva-

lent to saying 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:

Rule 4: Series with Independence. 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.

Symbolically (where n is the number of independent trials):

Pr(h at least once in n trials) 5 1 2 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.

Permutations and Combinations

Another common confusion is between permutations and combinations. A

permutation is a set of items whose order is specified. A combination is a set of

items whose order is not specified. Imagine, for example, that three cards—

the jack, queen, and king of spades—are facedown in front of you. If you

pick two of these cards in turn, there are three possible combinations: jack

and queen, jack and king, and queen and king. In contrast, there are six pos-

sible permutations: jack then queen, queen then jack, jack then king, king

then jack, queen then king, and king then queen.

Rule 2 is used to calculate probabilities of permutations—of conjunctions

of events in a particular order. For example, if you flip a fair coin twice, what

is the probability of its coming up heads and tails in that order (that is, heads

on the first flip and tails on the second flip)? Since the flips are independent,

Rule 2 tells us that the answer is 1/2 3 1/2 5 1/4. This answer is easily con-

firmed by counting the possible permutations (heads then heads, heads then

tails, tails then heads, tails then tails). Only one of these four permutations

(heads then tails) is a favorable outcome.

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