Simulating a die roll by flipping two coins

نویسنده

Giovanni Viglietta

چکیده

We show how to simulate a roll of a fair n-sided die by one flip of a biased coin with probability 1/n of coming up heads, followed by 3⌊log 2 n⌋+ 1 flips of a fair coin. Let a p-coin be a biased coin with probability p of coming up heads. A fair coin is a (1/2)-coin. A fair n-sided die is a die with n sides, each of which comes up with probability 1/n when the die is rolled. Proposition 1. For every positive integer n, and k = ⌊log2 n⌋, there are two integers a and b, with 0 6 a, b 6 2, such that a+ b(n − 1) = 2. Proof. If n = 2, we let a = b = 2. Otherwise 2 < n < 2, and we let a and b be, respectively, the remainder and the quotient of the Euclidean division of 2 by n− 1. Hence 0 6 a < n− 1 < 2, b > 0, and b = 2 − a n− 1 6 2 2 = 2. Lemma 2. A flip of a (2/n)-coin, where k = ⌊log2 n⌋, can be simulated by one flip of a (1/n)-coin and k + 1 flips of a fair coin. Proof. We show an algorithm that, given the outcomes of the coin flips, outputs “heads” with probability 2/n and “tails” with probability 1−2/n. Let a and b be as in Proposition 1, and let d < 2 be the non-negative integer whose i-th binary digit is 1 if and only if the fair coin comes up heads on the i-th flip, with 1 6 i 6 k + 1. We output “heads” if and only if the (1/n)-coin comes up heads and d < a, or the (1/n)-coin comes up tails and d < b. This happens with probability 1 n · a 2 + n− 1 n · b 2 = 2 n · 2 = 2 n . ∗School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa ON, Canada, [email protected].

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