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Fun new probability puzzle

Posted By: Timothy Chow
Date: Saturday, 27 May 2017, at 10:44 p.m.

In Response To: Fun new probability puzzle (Jeremy Bagai)

Bob Koca's comment about convexity can be visualized using this graph.

The curved line is the parabola y = x2, which enters the picture because when you do two independent trials of the same experiment, the probability of succeeding both times is the square of the probability of succeeding once. The two points where the diagonal line intersects the parabola are (1/2,1/4) and (5/6,25/36), indicating the probabilities of survival if you pick the 3-bullet or 1-bullet revolvers respectively; so your overall probability of survival if you pick one of them randomly is the average of these, which is indicated by the point where the vertical red line intersects the diagonal line.

On the other hand, if you pick the 2-bullet revolver, then your probability of survival is indicated by the point where the vertical red line intersects the parabola, which is lower.

Of course if we replace y = x2 with y = x3 or some higher exponent, the curve remains convex (or "concave up" if you prefer) so you're still better off picking one of the two guns at random.

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