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bonanova

six dice = four numbers

Question

Someone is willing to bet, at even odds, that if he throws six fair dice, exactly four distinct numbers will show.

For example 3 3 4 4 5 6 or 3 3 3 4 5 6.

Do you take the bet?

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There are two ways to get exactly four numbers: aaabcd and aabbcd.


First you can get in 6_choose_3*6*5*4*3 ways, and
second you can get in 6_choose_4*3*6*5*4*3 ways,
making total of (6_choose_3+6_choose_4*3)*6*5*4*3 = (20+15*3)*6*5*4*3 = 23400 ways.
We should not accept the bet if 23400/6^6 > 1/2.
It's close, but we are slight underdog, so we reject.

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