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Subject: Re: Probability problem  Date: 9/14/2020 8:58 PM  
Author: rev2217  Number: 6007 of 6013  
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 sixsided die, a 1/3 probability of choosing the 12sided die, and a 1/3 probability of choosing the 20sided die. But here's the catch: there are 38 possible outcomes of the first roll  1 through 6 on the 6sided die, 1 through 12 on the twelvesided die, and 1 through 20 on the twentysided 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 sixsided die is 1/18 (1/3 x 1/6), the probability of each outcome on the twelvesided die is 1/36 (1/3 x 1/12), and the probability of each outcome on the twentysided 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 sixsided die, the other eleven outcomes on the twelvesided die, and the other nineteen outcomes on the twentysized 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 sixsided die 10/18 = 5/9 of the time, from the twelvesided die 10/36 = 5/18 of the time, and from the twentysided 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 sixsided die because the sixsided 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 twelvesided die 5/8 of the time and from the twentysided die 3/8 of the time. And an initial roll of 13 or higher could come only from the 20sided die. Norm. 

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