Classic Problems of Probability

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

After an infinitely large ...