Rolling Coins

by Douglas Zare
2 August 2013

Douglas Zare

Suppose you are ready to play backgammon, but your dice are missing. You do have a coin. Can you use a sequence of coin tosses as a substitute for the dice?

The most efficient way to generate rolls with a coin may be to go buy some dice with the coin. However, if that's not possible, there are a few ways you can "create a roll" from a coin by tossing the coin and assigning some sequences to 1, some to 2, and some to 6.

You could flip the coin 5 times, and count the number of heads plus 1. This will give you a number from 1 to 6. However, this is bad because a 1 or 6 will be less likely than a 3 or 4. To get a 1, you would need to get 5 tails in a row, which happens with probability 1/32 instead of 1/6. We are only interested in creating fair dice, so we want the probability of each number from 1-6 to be 1/6.

No dice. Now what?

Here is a method which produces a fair roll of one die. Toss the coin 6 times. If there is only one head, say THTTTT, the roll is the location of the head, which would be 2 in this example. If there is more than one head, or if there are no heads, then toss the coins again.
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