Question

1. The hats of n persons are so that every assignment of the hats to the persons is equally likely). What is the probability thrown into a box. The persons then pick up their hats at random (i.e., that (a) every person gets his or her hat back? (b) the first m persons who picked hats get their own hats back? (c) everyone among the first m persons to pick up the hats gets back a hat belonging to one of the last m persons to pick up the hats? Now assume, in addition, that every hat thrown into the box has probability p of getting dirty (independently of what happens to the other hats or who has dropped or picked it up). What is the probability that (d) the first m persons will pick up clean hats? (e) exactly m persons will pick up clean hats?

Answer 1

1 a) n hats can be distributed in n! The probability of each getting own hat is n! m)! b) When m persons get their own hat, the remaining n - m hats can be distributed in ways. So the required probability is (n - т)! n! c) The hats of last m persons can be distributed in m! ways. The remaining n m hats can be distributed among the rest n m persons in -m)!ways. So the required probability is |(n -m)! 1 1 m! n! n! тI(n-m)! d) The probability of picking up a clean hat is P. The probability, m persons pick up clean hats is (1 - p)m e) Use binomial distribution. The probability of picking up a clean hat is-P. The probability n (1 -p)m p m n-m exactly m persons pick up clean hats is