| An infinite color spectrum in a bottomless pit |
There are x beads in a bottomless pit. Only two of them are the same color. Two beads are chosen at random. Let p(x) equal the probability that these two beads are the same color. Find
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Problem Moderated by: Sasha |
| Problem Solution |
There are x(x-1) different ways to draw two beads and two ways to draw the same color twice (since order matters). The probability function, therefore, equals 2/x(x-1), or equivalently, 2/x-1 - 2/x, which becomes a friendly telescoping sum as we take it from three to infinity.
2(1/2 - 1/3) + 2(1/3 - 1/4) + 2(1/4 - 1/5) + ... + 2(1/(n-1) - 1/n) + ...
Every term after 2(1/2) cancels out, and since the probability approaches zeros as the number of beads approaches infinity, the sum equals 1.
This summation was suggested by MasterShin. |
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