Parrando’s Paradox

One of the most interesting math ideas I’ve ever come across is called Parrando’s Paradox. Briefly, you have two games that, if played individually, will cause you to lose slowly but surely. However, by playing the games together in a certain sequence, you can actually win over time.

ParrandosParadox

Suppose the first game is to flip a biased coin. You win $1 if the result is Heads, but lose $1 if the result is Tails. The coin is biased so that the probability of Heads is 0.495 and the probability of Tails is 0.505 so you will slowly but surely lose over time.

In the second game, you take turns spinning one of two spinners, spinner A and spinner B. Spinner A is very bad and you have a probability of winning of only 0.095. Spinner B is very good and you have a probability of winning of 0.745.

You pick and spin spinner A if the amount of money you have is an even multiple of 3 ($3, $6, $9, etc.) You pick and spin spinner B if the amount of money you have is not an even multiple of 3.

Although it’s not entirely obvious, if you play this second game, you will slowly but surely lose money. About one-third of the time you’ll play spinner A and win 0.095 of the time, and about two-thirds of the time you’ll play spinner B and win 0.745 of the time. The approximate expectation per play is (1/3)(0.095)(+1) + (1/3)(0.905)(-1) + (2/3)(0.745)(+1) + (2/3)(0.255)(-1). If you do the math, it turns out your expectation is about -0.007 per play.

Now, suppose you take turns, playing G1 then G2 then G1 then G2, and so on. Well, if you play two losing games you will lose.

However, if you alternate by playing G1, G1, G2, G2, G1, G1, G2, G2, and so on, astonishingly, you will slowly win over time. Two losing games combine to give a winning game. Remarkable.

It isn’t really a paradox once you understand the tricky math involved. The Wikipedia article on Parrando’s Paradox explains what’s happening.

During my lunch break today, I wrote a little simulation program to verify the effect.

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