Expected attempts until the first made shot

Question

A player shoots free throws, making each one independently with probability 	frac14. They keep shooting until the first make. What is the expected number of attempts until the first made shot?

Accepted Answer

Let E be the expected number of attempts. The first attempt always happens. With probability 	frac14 it goes in and they stop; with probability 	frac34 it misses and they are back to the start: E = 1 + 	frac34 E \quad\Longrightarrow\quad 	frac14 E = 1 \quad\Longrightarrow\quad E = 4. Takeaway. With make probability p = 	frac14, the expected number of attempts to the first make is 1/p = 4.

Expected attempts until the first made shot

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