Question

You are in a room with 5 doors, and the homeowner tells you that behind two randomly chosen doors there is a prize (all doors are equally likely to be selected for a prize).
You open the doors, one by one, from left to right. Let X be the random variable that measures the number of doors you open after discovering the first prize, but before discovering the second prize.

A) What are the possible values ​​that X can take?
B) Determine the pmf of X
C) Determine the probability that X is at least 2

Answer 1

A)

You need to at least one door to open to get the prize.

The possible values of X are 1, 2, 3, 4 or 5.

B)

Probability to get prize on kth attempt = 1/(5-k+1) where k = 1, 2, ..., 5

Probability to not get prize on kth attempt = 1 - 1/(5-k+1) = (5-k)/(5-k+1) where k = 1, 2, ..., 5

The PMF of X is,

P(X = k) = Probability to fail to get prize by k-1 doors * Probability to get prize on kth door

for  = 1, 2, ..., 5

  for  = 1, 2, ..., 5

C)