Chance a point appears at least once in a bootstrap sample

Question

You have n = 200 observations and draw a bootstrap sample of size 200 with replacement. What is the probability that one particular observation is selected at least once (i.e. it appears in the bootstrap sample)?

Accepted Answer

The point appears at least once unless it is missed on all n independent draws. The miss-everything probability is \left(1 - 	frac{1}{n}ight)^{n}, so the "appears" probability is its complement. Probability of inclusion. 1 - \left(1 - \frac{1}{n}ight)^{n} = 1 - \left(1 - \frac{1}{200}ight)^{200} \approx 1 - 0.367 = 0.633. As n grows this converges to 1 - e^{-1} \approx 0.632. So each observation has about a 63\% chance of being represented in any given bootstrap sample, and about a 37\% chance of sitting out as an out-of-bag point. Takeaway. Each observation makes it into a bootstrap r…

Chance a point appears at least once in a bootstrap sample

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