A blind man is on a strange island and he has 2 red pills and 2 white pills, completely identical and has kept in his pockets, he needs to take 1 red pill and 1 white pill order doesn't matter. If he takes 2 pills of the same color he dies. How does he survive. There is a standard solution for this and I was wondering if there could be a combinatorial way to solve this?
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Chances that he survives $=1-$Chances that he dies
$1-\dfrac{2}{^4C_2}=1-\dfrac{2}{6}=\dfrac{2}{3}=66.67$% Chances
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I'm not sure what you would consider combinatorics, and what you wouldn't. The standard solution mentioned in other comments seems combinatorial to me. The insight necessary is that the pills are not "atomic", they can be divided.
One non-standard way to solve, which is kind of the same, is to smash the pills into infinitesimally small bits and then blend them until the particle distribution is uniform and well mixed, then eat half of the pile. with probability 1, you win.
The fact that doing this subdivision and recombination in the most coarse possible way ( break in two and recombine any way you like, then eat half ) actually works with probability 1 as well seems like a numerical miracle, which is kind of what I like to think combinatorics is.
Just take a glass of water, fill it to the brim. Put all the pills inside, they should dissolve well, all pills will dissolve well, drink half the glass of water, even a blind man can tell, problem solved.