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## Thursday, December 22, 2011

### the odds of 3 (or more) dice add-up

A lot of probability puzzles use small integers such as dice, coins, fingers, or poker cards. These puzzles are usually tractable until we encounter the *sum* of several dice.

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter7.pdf has a concise treatment on the probability when adding up 2 dice or 3 dice or 4 .....

The same _thought_process_ can help solve the probability of X + 2Y + 3Z, where X/Y/Z are integers between 1 and 10, i.e. fingers.

Basically the trick is to construct the probability function of 2 dice combined. Then use that function to work out the prob function of 3 dice, then 4 dice.... Mathematical induction.

Compared to integers, continuous random variables are harder, where the "dice" can give non-integers. Prob of sum of 2 continuous random variables requires integration. While clever people can avoid integration, 3 (uniform, independent) variables are perhaps tricky without integration. But I don't feel many probability puzzles ask about 3 random real numbers summing up.

As a layman, I feel the best aid to the "continuous" probability is the integer result above.