probability of rolling 6 dice

Question

what is the probability of rolling 6 dice at once, where you'll get exactly 2 of the same pair out of 6 dice? Each die is a 6 sided die. numbering 1-6.

Answer 1

So Nick S.,

The trick to doing these problems is to frame them with just the right amount of specificity.

So, roll the first die. You don't care what it shows, it shows some number. OK. That was probability 1 !.

Roll the second die: what is the probability it matches the first? Exactly 1/6 . OK, that gives two dice., P(X,X)=1*1/6=1/6.

Roll the 3rd die: what is the probability that it doesn't match #1 (=#2) ? it's 5/6

The 4th, P not-matching either 1/2 or 3 = 4/6

The 5th, P not matching either 1/2,3, or 4 = 3/6

The 6th, P not matching either 1/2, 3, 4, or 5 = 2/6

So the conditional string of probability so far is (1*5*4*3*2)/6^5 for just the 1/2 pair ~0.015.

Do we care that there are several different orders in which the pair could have been rolled? Not at all! We are concerned with probability of the exact result, two dice match, and the rest do not. We could mix them around all we like, the probability of having rolled the overall result wouldn't change!

So what did I mean about right amount of specificity? Here, it meant: don't specify the NUMBER rolled for 1&2. And don't fret about the ORDER the dice were rolled in. The number of permutations is irrelevant. The dice could have been rolled in any order, the outcome (as defined in this problem) is unchanged!

-- Cheers, --Mr. d.