# Random forest - Out-of-bag estimates

I am reading the chapter on random forests by Leo Breiman (found here: https://www.stat.berkeley.edu/~breiman/randomforest2001.pdf).

In section 3.1 Using out-of-bag estimates to monitor error, strength, and correlation (page 11), it says:

In each bootstrap training set, about one-third of the instances are left out. Therefore, the out-of-bag estimates are based on combining only about one-third as many classifiers as in the ongoing main combination.

I am not sure I understand how the first sentence (that about one-third of cases are left out of each bootstrap sample) implies the second (that each case is OOB in about one-third of the trees)?

• Check if the definitions in this paper can help you: arxiv.org/abs/2112.06101
– Zen
Commented Dec 14, 2021 at 2:56

In a bootstrap sample we replace. The probability of a subject $$x$$ being sampled with replacement is at about $$\frac{2}{3}$$. If we build for example $$1000$$ trees with different bootstrap samples, we expect that $$x$$ will be in $$\frac{2}{3} * 1000$$ of these samples. We expect that one third of the trees were therefore not build with subject $$x$$. The OOB only calculates the error for the trees that have not been build with $$x$$, which is $$\frac{1}{3} * 1000$$ in that case.