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Classic Problems of Probability by Prakash Gorroochurn

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Problem 33

Parrondo's Perplexing Paradox (1996)

Problem. Consider the following games G1, G2, and G3, in each of which $1 is won if a head is obtained, otherwise $1 is lost (assuming the player starts with $0).

G1: A biased coin that has probability of heads .495 is tossed.

G2: If the net gain is a multiple of three, coin A is tossed. The latter has probability of heads .095. If the net gain is not a multiple of three, coin B is tossed. The latter has probability of heads .745.

G3: G1 and G2 are played in any random order.

Prove that although G1 and G2 each result in a net expected loss, G3 results in a net expected gain.

Solution. For game G1 the net expected gain is $1(.495) − $1(.505) = −$.01.

For game G2, let img denote the net gain of the player after n tosses. Then, given all the prior values of the net gain, the value of img depends only on img, that is, img is a Markov chain.1 Moreover, img2 is also a Markov chain with states {0, 1, 2} and transition probability matrix

After an infinitely large ...

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