You are working in a bank and you are serving customers who

You are working in a bank and you are serving customers who are making deposits. When you serve 100 customers, your shift ends and you go home. Each minute a customer arrives at the bank, and either makes a deposit with probability 1/3. or makes a withdraw with probability 2/3 (independently of all other customers). If it happens that three consecutive customers make a withdraw, you get bored and fall asleep, and you stay-asleep until a customer making deposit arrives (and awakes you). Let T be the number of minutes you spend sleeping at work before you go home. Find the expectation of T.

Solution

here T follow geometric distribution with p=p/3, q=1/3

The expectation of T is given by

E(T)=q/p=(1/3)/(2/3)=1/2