Question

You are at a campus party where there are a total number of n people. The host asked everyone to put their phones in a bowl while walking in. A noise complaint ends the party abruptly, and everyone heads for the door, hastily grabbing their phones from the bowl Assume every guest has one and exactly one phone, and that they pick a phone at random (so that every assignment of a phone to a person is equally likely). What is the probability that: a. Every person gets their phone back? b. The first m persons to pick each get their own phones back? c. The first m persons to pick each get a phone belonging to the last m persons to pick? Hint: Try this thought experiment with a few choices of mand n to get a feel for the numbers that show up.)

Answer 1

1,
`(m)/(n)`,
`(1-m)/(n)`.

Explanation:

probability =
`(Number of Possible Outcomes)/(Total Outcomes)`

Total number of persons in the party = n

a) Pr ( every person gets their phone back) = Pr (each person picks his phone ) multiplied by number of person

=
`(1)/(n)` × n = 1.

No of first m persons to pick = m

No of last m persons to pick = 1 - m

b) Pr (first m persons to pick each gets their phones back) =
`(m)/(n)`

c) Pr( first m persons get a phone belonging to last m persons) =
`(1-m)/(n)`