This problem uses martingales to find the expected number of trials E [J] before a fixed pattern, a 1 , a 2 , . . . , a k , of binary digits occurs within a sequence of IID binary random variables X 1...

This problem uses martingales to find the expected number of trials E [J] before a fixed pattern, a₁, a₂, . . . , a_k, of binary digits occurs within a sequence of IID binary random variables X₁, X₂, . . . A mythical casino and set of gamblers who follow a prescribed strategy will be used to determine E [J]. The casino has a game where, on the ith trial, gamblers bet money on either 1 or 0. After bets are placed, Xi above is used to select the outcome 0 or 1. Let p(1) = pX(1) and p(0) = 1 p(1) = pX(0). If an amount s is bet on 1, the casino receives s if X_i
= 0, and pays out s/p(1) s (plus returning the bet s) if X_i
= 1. If s is bet on 0, the casino receives s if X_i
= 1, and pays out s/p(0) s (plus the bet s) if X_i
= 0.