(a) Here the sample space is all the negative and positive integers including zero i.e. S = {...,-3,-2,-1,0,1,2,3,...}
(b) P[He wins at jth spin] = P[He lose previous j-1 spins and He wins at jth spin]
= P[He lose previous j-1 spins] * P[He win at jth spin] {Both the event is independent}
=
(c) Yes, the game is not fair for him. The chance of winning is decreasing as he plays more. For each spin, the probability of winning is not the same as the probability of loss i.e. not equal to 1/2.
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