My Favourite Martingale Problems

ABRACADABRA

At each of times 1,2,3,…., a monkey types a capital letter at random, the sequence of letters typed forming an iid sequence of random variables each chosen uniformly from amongst the 26 possible capital letters.

Just before each time n=1,2,…., a new gambler arrives on the scene, He bets $1that

\text{the $n^{th}$ letter will be A.}

If he loses, he leaves. If he wins, he receives $26 all of which he bets on the event that

\text{the $(n+1)^{th}$ letter will be B.}

If he loses, he leaves. If he wins, he receives $26 all of which he bets on the event that

\text{the $(n+2)^{th}$ letter will be R.}

and so on through the ABRACADABRA sequence. Let T be the first time by which the monkey has produced the consecutive sequence ABRACADABRA. Explain why martingale theory makes it intuitively obvious that

E(T) = 26^{11}+26^4+26

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