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Classic Problems of Probability by Prakash Gorroochurn

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Problem 29

Lévy and the Arc Sine Law (1939)

Problem. A fair coin is tossed successively. Let img if the jth toss is a head, and img if the jth toss is a tail. Define the difference between the number of heads and tails after j tosses by img. Show that the probability that the last time there was an equal number of heads and tails within 2n tosses occurred at the 2kth toss is img.

Solution. Let img, where img is the number of ways the event img can be realized. We first establish an important result. We have

(29.1) equation

since the coin is fair. Using the law of total probability,

img

Applying the Ballot Theorem (see Problem ...

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