Omission probability for a tiny bootstrap sample

Question

You have just n = 5 observations and draw a bootstrap sample of size 5 with replacement. What is the exact probability that one particular observation is left out of the bootstrap sample entirely?

Accepted Answer

On each draw the specific point is missed with probability 1 - 	frac{1}{5} = 	frac{4}{5}. With five independent draws, omission requires missing on all five. Probability of omission. \left(\frac{4}{5}ight)^{5} = \frac{1024}{3125} \approx 0.328. Compare this with the large-sample limit e^{-1} \approx 0.368: at n = 5 the omission probability is a bit lower. The value \left(1 - 	frac{1}{n}ight)^n climbs toward 1/e from below as n grows, so small samples leave out slightly fewer points than the asymptotic rule of thumb suggests. Takeaway. The 37\% out-of-bag figure is a large-n limit; for …

Omission probability for a tiny bootstrap sample

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