Mathematical Methods in the Physical Sciences · Problem 4.14
FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas
Mathematics — Methods · probability-all-faces-die
Problem
(a) Find the probability that in two tosses of a coin, one is heads and one is tails. That in six tosses of a die, all six of the faces show up. That in 12 tosses of a 12-sided die, all 12 faces show up. That in n tosses of an n-sided die, all n faces show up.
(b) The last problem in (a) is equivalent to finding the probability that, when n balls are distributed at random into n boxes, each box contains exactly one ball. Show that for large n, this is approximately e−n2πn.