Expected guesses to hit a secret digit

Question

A secret single digit from 0 to 9 is fixed. You guess digits at random, each guess correct with probability $\tfrac{1}{10}$ independently, stopping when you guess it right.

What is the expected number of guesses until you are correct?

Accepted Answer

Let E be the expected number of guesses. The first guess always happens. With probability 	frac{1}{10} it is right and you stop; with probability 	frac{9}{10} it is wrong and you are back to the start: E = 1 + 	frac{9}{10}E \quad\Longrightarrow\quad 	frac{1}{10}E = 1 \quad\Longrightarrow\quad E = 10. Takeaway. With success probability p = 	frac{1}{10}, the expected wait is 1/p = 10 guesses.

Expected guesses to hit a secret digit

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