**An Example: Coin-tossing**

The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. With a fair coin, the outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is exactly 1⁄_{2} (one in two). It follows that the probability of getting two heads in two tosses is 1⁄_{4} (one in four) and the probability of getting three heads in three tosses is 1⁄_{8} (one in eight). In general, if we let *A _{i}* be the event that toss

*i*of a fair coin comes up heads, then we have,

- .

Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is only 1⁄_{32} (one in thirty-two), a believer in the gambler's fallacy might believe that this next flip is less likely to be heads than to be tails. However, this is not correct, and is a manifestation of the gambler's fallacy; the event of 5 heads in a row and the event of "first 4 heads, then a tails" are equally likely, each having probability 1⁄_{32}. Given the first four rolls turn up heads, the probability that the next toss is a head is in fact,

- .

While a run of five heads is only 1⁄_{32} = 0.03125, it is only that *before* the coin is first tossed. *After* the first four tosses the results are no longer unknown, so their probabilities are 1. Reasoning that it is more likely that the next toss will be a tail than a head due to the past tosses, that a run of luck in the past somehow influences the odds in the future, is the fallacy.

Read more about this topic: Gambler's Fallacy