Problem 8

Jacob Bernoulli and His Golden Theorem (1713)

Problem. Show that the probability of an equal number of heads and tails when a fair coin is tossed 2m times is approximately for large m.

Solution. Let X be the number of heads for 2m independent tosses of a fair coin. Then X B(2m, 1/2) and

The probability of an equal number of heads and tails is

Applying Stirling's formula for large N, we have

(8.1)

# 8.1 Discussion

The last formula implies that, for a fair coin, the probability of an equal number of heads and tails is inversely proportional to . Table 8.1 shows that, as more coins are tossed, it becomes less likely that there will be an equal number of heads and tails.

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