Suppose that I toss a sequence of coins that come up heads with probability p. On average, how many coins do I have to toss before I see a heads?

Marlene Brooks 2022-10-12 Answered
Recursive proof of expectation of geometric distribution?
If T G e o ( p ) ( 0 < p < 1) then E T = 1 / p is well known.
Suppose that I toss a sequence of coins that come up heads with probability p. On average, how many coins do I have to toss before I see a heads?
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Answers (1)

Jovanni Salinas
Answered 2022-10-13 Author has 18 answers
Step 1
We can condition on the first toss, let H denote the event that the first toss is a head.
E [ T ] = E [ T | H ] E [ H ] + E [ T | H c ] E [ H c ] = p + ( 1 + E [ T ] ) ( 1 p ) = 1 + ( 1 p ) E [ T ]
Step 2
Solving for E[T] gives us 1 p .
Notice that E [ T | H c ] = 1 + E [ T ] as the first toss and the remaining tosses are independent.
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