EXAMPLE 21 Gambler's Fallacy

Suppose we have tossed a fair coin 10 times and have observed heads come up every time. Find the probability of tails on the next toss.

Solution

We have observed an unusual number of heads, so we might think that the probability of tails on the next toss is increased. However, the short answer is “Not so.” Successive tosses of a fair coin are independent because the coin has no memory of its previous tosses. Thus, what happened on the first 10 tosses has no effect on the next toss. Probability theory tells us that, in the long run, the proportion of heads and tails will eventually even out if the coin is fair. Therefore, the probability of tails on the next toss is 0.5. This is an example of the Gambler’s Fallacy.