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Aha! Bayesian analysis makes it clear.

Posted By: Timothy Chow
Date: Sunday, 28 May 2017, at 2:59 p.m.

In Response To: Aha! Bayesian analysis makes it clear. (Jeremy Bagai)

Here's another way of thinking about it, that addresses your question about variance.

Imagine firing both times even if there is a bullet in the first round, and let B denote the number of bullets that get fired (so B = 0, 1, or 2).

In both cases, the expected value of B is the same, but in the 1-or-3 case, the variance of B is higher.

What you're actually interested in is the probability that B = 0; i.e., you're interested in a so-called "tail probability." Generally speaking, of two random variables with the same mean, the one with the higher variance will have the higher tail probability.

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