Gacha pulls until the first rare item

Question

In a game, each pull independently yields the rare item with probability p = 0.02. Let N be the number of pulls up to and including the first rare item. What are the mean and variance of N?

Accepted Answer

N is geometric (trials until the first success), so mean = 1/p and variance = (1-p)/p^2. With p = 0.02: \mathbb{E}[N] = \frac{1}{0.02} = 50, \qquad \operatorname{Var}(N) = \frac{0.98}{0.02^2} = \frac{0.98}{0.0004} = 2450. Standard deviation \sqrt{2450} \approx 49.5, almost as large as the mean itself. With such low odds, the number of pulls needed swings wildly: some players get lucky fast, others grind far past 50. Takeaway. At p = 0.02, expect 50 pulls with variance 2450; when success is rare, the standard deviation is nearly equal to the mean.

Gacha pulls until the first rare item

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