Question

A key chain holds N different keys (N ≥ 2) and only one of them opens the door to your apartment. You try the keys on the key chain at random until the door unlocks. Let X be the number of times you tried unlocking the door. What is the probability mass function of X and the mean value of X when:

(a) After trying a key you leave it on the key chain.

(b) After trying a key you remove it from the key chain.

Answer 1

Solution

X follows geometric distribution with p = 1/n, 1-p = n-1/n

(a) E(X) = 1/p = 1/1/n = n

(b) P(X = 1) = 1/n

P(X= 2) = ((n-1)/n)(1/(n-1)) = 1/n

P(X=3) = ((n-1)/n)(n-2/n-1)(1/n-2) = 1/n

……..P(X= n ) = 1/n

E(X) = 1/n + 2/n + 3/n+.....+n/n = n(n+1)/2n = (n+1)/2