given X1,X2,...,Xn (zeta replaced by x for convenience) follow Bernoulli distribution with p = 7/8 with
therefore Sn follows binomial distribution(n,p=7/8).
E(Sn/n) = np/n = p =7/8
Var(Sn) = pq/n =(7/8*1/8)/n
choosing k =0.1,
we have
1-(pq/n)/(0.1*0.1) =0.8
n =7/0.128
n = 54.6875
hence n = 55
so he has to play at least n = 55 games to get gain = np = 55*(7/8) =48.125.
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