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3. (Let's look back and solve an interesting counting problem) The hats of n persons are thrown into a box. The persons then pick up their hats at random (i.e., so that every assignment of the hats to the persons is equally likely). What is the probability 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

3. (Let's look back and solve an interesting counting problem) The hats of n persons are thrown into a box. The persons then pick up their hats at random (i.e., so that every assignment of the hats to the persons is equally likely). What is the probability 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

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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3. (Let's look back and solve an interesting counting problem) The hats of n persons are thrown into a box. The persons then pick up their hats at random (i.e., so that every assignment of the hats to the persons is equally likely). What is the probability 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?
Transcribed Image Text:3. (Let's look back and solve an interesting counting problem) The hats of n persons are thrown into a box. The persons then pick up their hats at random (i.e., so that every assignment of the hats to the persons is equally likely). What is the probability 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?
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Step 1: Given Information

The problem statement provides the following information:- 

There are n people and n hats.

The hats are thrown into a box and picked up at random by the person.

Each assignment of the hat to the person is equally likely. 

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