1. p people, each wearing a different hat, throw their hats into the ring. They all have second t...
1. p people, each wearing a different hat, throw their hats into the ring. They all have second thoughts, run into the ring, and start randomly grabbing as many hats as they can, without regard to ownership, some obtaining many hats, others emerging from the chaos hatless. Compute the probability that every person who manages to grab at least one hat manages to retrieve not only their own hat but also one or more hats that do not belong to him or her. Make separate counting arguments for (a) the denominator of the probability expression and (b) the numerator of the probability expression. The denominator has a fairly simple expression, but the numerator should be expressed in terms of Stirling numbers of the second kind.
1. p people, each wearing a different hat, throw their hats into the ring. They all have second thoughts, run into the ring, and start randomly grabbing as many hats as they can, without regard to ownership, some obtaining many hats, others emerging from the chaos hatless. Compute the probability that every person who manages to grab at least one hat manages to retrieve not only their own hat but also one or more hats that do not belong to him or her. Make separate counting arguments for (a) the denominator of the probability expression and (b) the numerator of the probability expression. The denominator has a fairly simple expression, but the numerator should be expressed in terms of Stirling numbers of the second kind.