Question

John is playing a slot machine game on his smart phone, and pays $1.00 for each spin. He can

play one spin after another until he either quits or wins a game. If he quits, he loses all of the

money he paid for the spins. However, he will get all of his money back if he wins a spin. Let

Ai be the event that he wins at ith spin, and P(Ai) = 1/5^i Moreover, the events A_i’ s are

pairwise mutually exclusive.

(a) What is the sample space S?

(b) Calculate the probability that he wins a given spin.

(c) Is this game fair for him? Briefly explain.

Answer 1

(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 j^th spin] = P[He lose previous j-1 spins and He wins at j^th spin]

= P[He lose previous j-1 spins] * P[He win at j^th 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.