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

∞

Σp(x)

X=3

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An infinite color spectrum in a bottomless pit 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

x(x-1)

, or equivalently,

x-1

, 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.

Theorem of Pappus: Continued

* Operations and Matrices

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