You are working in a bank and you are serving customers who are making deposits.

Question

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.

Explanation / Answer

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

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