Rolling 3 Pairs with 6 Dice
Date: 12/19/2001 at 12:00:53 From: Jeremy Kuhn Subject: Probability of rolling 3 pairs when rolling 6 dice Upon playing a game - '10,000' on the computer I was struck by the frequency with which 3 pairs came up. After testing it came to around 6% of the time. Trying this with physical dice by hand I came up with a similar percentage. I tried to verify this mathematically and could not. I assumed that the sample would be 6^6 (6 dice with 6 different possibilities) and that the number of possibilities of 3 pairs would be 6^3 (divided the dice into pairs and figured there would be 3 pairs of dice with 6 different possibilities of being a match each). This then led to 216/46656 or approximately 0.5% chance of rolling 3 pairs. Where am I going wrong ? Thanks! Jeremy
Date: 12/21/2001 at 15:27:42 From: Doctor Jaffee Subject: Re: Probability of rolling 3 pairs when rolling 6 dice Hi Jeremy, To solve this problem, I would get the denominator the same way, 6^6, but I would look at the numerator differently. First of all, 6^3 implies that the second pair of numbers could be the same as the first pair, but then it wouldn't be a pair any more. So, I look at it this way. Suppose we pick three numbers; for example, 2,4,5. How many ways can we get pairs of these numbers in 6 rolls? In other words, we need to know how many ways we can create sequences like 2,4,4,5,2,5. If we place the 2's first, there are 6 slots to choose from, so the number of ways to place the 2's is as many as there are combinations of 6 taken 2 at a time, or 15. Then there are 4 slots left for the next pair which gives us "4 choose 2" or 6 choices. Finally, there are 2 slots left for the remaining pair. There is one way to do that. Therefore, the number of ways to roll a pair of 2's, a pair of 4's and a pair of 5's in 6 rolls is 15 x 6 = 90. But, if we are going to roll 3 pairs of 6 dice we have 3 numbers to choose from the 6, so there are actually 20 different groups of 3 numbers that we can use. Consequently, the total number of ways of rolling 3 pairs is 20 x 90 = 1800. Now 1800/6^6 is ony about 3.8%. Were you counting 4 of one number and two of another number, or 6 of one number as 3 pairs? If the latter, that raises the probability to about 4.35% Conceivably, your experimental result of about 6% could have been attributed to the fact that you were a little luckier than you should have been. Very likely, if you were to perform the experiment again you would get only 3% of the rolls generating 3 pairs of doubles. And, of course, over the long run, you should find that 3 pairs show up in 3.8 or 4.35% of the time, depending on how you are defining "three doubles." I hope this answer is helpful to you. I consulted with Dr. Anthony and Dr. Mittledorf and we all agreed. Write back if you want to discuss this problem some more. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
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