plj8624
04-16-2004, 02:00 AM
seemingly paradoxical problem posted from page 52:
Imagine you are standing on stair 0, in the middle of a very long staircase with 1,001 stairs numbered from -500 to 500 (-500, -499, -498... -4, -3, -2, -1, 0, 1, 2, 3, 4 ... 498, 499, 500). You want to go up rather than down the staircase and which direction you move depends on the outcome of coin flips. The first game- let's call it game S - is very simple. You flip a coin and move up a stair whenever it comes heads and down a stair whenever it comes up tails. The coin is slightly biased and comes up heads 49.5% of the time and tails 50.5% of the time. If you play this game you would almost certainly end up at the bottom of the staircase.
The second game- game C - is more complicated. It involves two coins, one of which, the bad one, comes up heads only 9.5% of the time, tails 90.5%. The other coin, the good one, comes up heads 74.5% of the time and tails the other 25.5%. As in game S, you move up a stair if the coin you flip comes up heads and you move down one if it comes up tails. But which coin do you flip? If the number of the stair you're on is a multiple of 3 (that is, ..., -9, -6, -3, 0, 3, 6, 9, 12, ...), you flip the bad coin. If the number of the stair you're on is not a multiple of 3, you flip the good coin. If you play this game over the long haul, chances are you will end up at the bottom of the staircase.
Parrondo's facinating discovery is that if you play these two games in succession in random order (keeping your place when you switch between games), you will steadily ascend to the top of the staircase. Alternatively, if you play two games of S followed by two games of C followed by two games of S and so on, all the while keeping your place on the staircase as you switch between games, you will also steadily rise to the top of the staircase.
Getting the Best of It is my favorite 2+2 book, but these finding seem to fly in the face of that book. Can anyone explain why my intuitive mathematical sense that randomly switching between two losing games can yield winning results?
Imagine you are standing on stair 0, in the middle of a very long staircase with 1,001 stairs numbered from -500 to 500 (-500, -499, -498... -4, -3, -2, -1, 0, 1, 2, 3, 4 ... 498, 499, 500). You want to go up rather than down the staircase and which direction you move depends on the outcome of coin flips. The first game- let's call it game S - is very simple. You flip a coin and move up a stair whenever it comes heads and down a stair whenever it comes up tails. The coin is slightly biased and comes up heads 49.5% of the time and tails 50.5% of the time. If you play this game you would almost certainly end up at the bottom of the staircase.
The second game- game C - is more complicated. It involves two coins, one of which, the bad one, comes up heads only 9.5% of the time, tails 90.5%. The other coin, the good one, comes up heads 74.5% of the time and tails the other 25.5%. As in game S, you move up a stair if the coin you flip comes up heads and you move down one if it comes up tails. But which coin do you flip? If the number of the stair you're on is a multiple of 3 (that is, ..., -9, -6, -3, 0, 3, 6, 9, 12, ...), you flip the bad coin. If the number of the stair you're on is not a multiple of 3, you flip the good coin. If you play this game over the long haul, chances are you will end up at the bottom of the staircase.
Parrondo's facinating discovery is that if you play these two games in succession in random order (keeping your place when you switch between games), you will steadily ascend to the top of the staircase. Alternatively, if you play two games of S followed by two games of C followed by two games of S and so on, all the while keeping your place on the staircase as you switch between games, you will also steadily rise to the top of the staircase.
Getting the Best of It is my favorite 2+2 book, but these finding seem to fly in the face of that book. Can anyone explain why my intuitive mathematical sense that randomly switching between two losing games can yield winning results?