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

to go home, the sum of deposits and withdrawals should be equal to 100.

Also, since three consecutive withdrawals make me sleepy,

the expectation/mean value of being sleepy=2/3*100*1/3=22.22