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/