Slot-machine pulls until the first jackpot

Question

A slot machine pays its jackpot on each pull independently with probability $1/50$ . A player keeps pulling the lever until the first jackpot.

On average, how many pulls does that take (counting the winning pull)?

Accepted Answer

Each pull is an independent trial with success probability p = 1/50. We want the expected number of pulls until the first jackpot. By the one-step argument, the first pull costs 1, succeeds with probability p, and otherwise resets the problem: E = 1 + (1-p)\,E \quad\Longrightarrow\quad E = \frac{1}{p} = \frac{1}{1/50} = 50 	ext{ pulls.} Notice how directly rarity maps to waiting: a 1-in-50 event takes 50 tries on average, no more and no less. Takeaway. Expected pulls to the first jackpot is 1/p = 50; the expected wait is simply the reciprocal of the per-try chance.

Slot-machine pulls until the first jackpot

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