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# Help me with some dice math, anyone?

edited February 2008
Say you've got three six-sided dice. Two of them look the same, while the third is special -- a different size, or color, or whatever. (Represented in this thread like this.)

You roll all three dice at once. Can any of you Masters of the Black Arts of Probability tell this innumerate moron the odds of the following results occurring?

1) The special die is one of the two highest dice (i.e., 6, 5, 3)?

2) The two highest dice are a pair, and the special die is one of them (i.e., 5, 5, 2)?

3) The two highest dice are a pair, and the special die is not one of them (i.e., 4, 4, 1)?

4) There's a pair, the special die is one of them, but another die is higher (i.e., 1, 1, 6)?

5) There's a pair, but the special die is both higher and not part of the pair (i.e., 2, 2, 5)?

• What about ties? In case 1, 2, 3, would "4, 4, 4" qualify?
• Assuming that words like "highest dice" can refer to any two dice that can be considered highest unless otherwise noted:

1) About 74.5% = (1*1 + 2*3 + 3*5 + 4*7 + 5*9 + 6*11) / 6^3

3) About 6.9% = (5 + 4 + 3 + 2 + 1 + 0) / 6^3