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Wayne,

Normally, when something is "picked at random" all valid items have the same chance to be picked.

That's right. There's a probability of 1/3 of initially choosing the six-sided die, a 1/3 probability of choosing the 12-sided die, and a 1/3 probability of choosing the 20-sided die.

But here's the catch: there are 38 possible outcomes of the first roll -- 1 through 6 on the 6-sided die, 1 through 12 on the twelve-sided die, and 1 through 20 on the twenty-sided die. The outcomes for each die are equiprobable, but that does not make all 38 outcomes equiprobable. In particular, the actual probability of each outcome on the six-sided die is 1/18 (1/3 x 1/6), the probability of each outcome on the twelve-sided die is 1/36 (1/3 x 1/12), and the probability of each outcome on the twenty-sided die is 1/60 (1/3 x 1/20).

Now, the knowledge that the initial roll is a 4 means that we have to throw out all outcomes that don't have a 4 as the rolled value -- that is, the other five outcomes on the six-sided die, the other eleven outcomes on the twelve-sided die, and the other nineteen outcomes on the twenty-sized die -- and scale the probabilities of the remaining cases (that is, the cases that do produce a four) so they add up to ONE. Now,

1/18 + 1/36 + 1/60 = 10/180 + 5/180 + 3/180 = 18/180 = 1/10

so we multiply the probabilities of the remaining cases by 10 -- and the initial roll of 4 came from the six-sided die 10/18 = 5/9 of the time, from the twelve-sided die 10/36 = 5/18 of the time, and from the twenty-sided die 10/60 = 1/6 of the time.

This is what's known as "conditional probability" -- the knowledge that the initial roll was a 4 alters the probability each die was chosen.

Note that the impact would have been very different if the initial roll had been 7 or more. If it were in the range of 7 through 12, the first die could not have been the six-sided die because the six-sided die does not have those values -- and 1/36 + 1/60 = 5/180 + 3/180 = 8/180 = 2/45, so it would come from the twelve-sided die 5/8 of the time and from the twenty-sided die 3/8 of the time. And an initial roll of 13 or higher could come only from the 20-sided die.

Norm.
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