Reference no: EM131834883

In a certain betting contest you may choose between two games A and B at the start of every turn. In game A you always toss the same coin, while in game B you toss either coin 1 or coin 2 depending on your bankroll. In game B you must toss coin 1 if your bankroll is a multiple of three; otherwise, you must toss coin 2. A toss of the coin from game A will land heads with a probability of ½ - ? and tails with a probability of ½ + ?, where ? = 0.005. Coin 1 in game B will land heads with probability 1/10 - ? and tails with probability 9/10 + ?; coin 2 in game B will land heads with probability 3/4 - ? and tails with probability ¼ + ?. In each of the games A and B, you win one dollar if heads is thrown and you lose one dollar if tails is thrown. An unlimited sequence of bets is made in which you may continue to play even if your bankroll is negative (a negative bankroll corresponds to debt). Following the strategy A, A,..., you win an average of 49.5% of the bets over the long-term. Use computer simulation to verify that using strategy B, B,..., you will win an average of 49.6% of the bets over the long-term, but that using strategy A, A, B, B, A, A, B, B,... you will win 50.7% of the bets over the long term. (The paradoxical phenomenon, that in special betting situations winning combinations can be made up of individually losing bets, is called Parrondo's paradox after the Spanish physicist Juan Parrondo, see also G.P. Harmer and D. Abbott, "Losing strategies can win by Parrondo's paradox," Nature, 402, 23/30 December 1999. An explanation of the paradox lies in the dependency between the betting outcomes. Unfortunately, such a dependency is absent in casino games.)