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Classic Problems of Probability by Prakash Gorroochurn

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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 img for large m.

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

img

The probability of an equal number of heads and tails is

img

Applying Stirling's formula img for large N, we have

(8.1) equation

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 img. 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.

Table 8.1 The Probability of an Equal Number of Heads and Tails When a Fair Coin is Tossed 2m Times.

Number ...

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