Problem 29
Lévy and the Arc Sine Law (1939)
Problem. A fair coin is tossed successively. Let 
if the jth toss is a head, and 
if the jth toss is a tail. Define the difference between the number of heads and tails after j tosses by 
. 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 
.
Solution. Let 
, where 
is the number of ways the event 
can be realized. We first establish an important result. We have
since the coin is fair. Using the law of total probability,
Applying the Ballot Theorem (see Problem ...
