We flip a coin until a tail or five heads in a row occur. What is the number of expected flips?

Question

We flip a coin until a taild or five heads in a row occur. What is the number of expected flips?

I have tried to solve this by first defining 2 random variables: X which is 1/2 if head and 1/2 if tail. Then I build the distribution table for this random variable. I don't know exactly how to define the second radom variable.

How would you approach this problem?

Answer 1

If I understand the problem correctly, there are only six possible sequences:

T; HT; HHT; HHHT; HHHHT; or HHHHH.

It should be simple enough to build a probability distribution for the number of flips until you are done, and then find the expected value.

Answer 2

Hint: Simpler to look at the possible events. At each flip, Heads or Tails may occur. By the fifth flip, the game is over.

Can you work out what is the probability of Flip 1 ending the game? Or Flip 2? etc.