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thomblake comments on An Intuitive Explanation of Solomonoff Induction - Less Wrong
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That doesn't work. If the probability of a regular soldier being able to hit the helmet once is p, then conditional on him hitting the helmet the first time, the probability that he can hit it again is still p. Your argument is the Gambler's fallacy.
The interesting bit is that the helmet was hit twice, so we're looking at the probability of being shot twice, not the probability of being shot the second time conditional on being shot the first time.
In retrospect, my first attempt at explanation was fairly poor. Is this clearer?
Edit: To more specifically address your objection: P(hit twice) = P(hit 1st time) * P(hit 2nd time | hit 1st time) = P(hit)^2.
yes