# Dice probability statistics pondering

• Jan 17th 2011, 09:40 PM
AshleyT
Dice probability statistics pondering
Okey, I'm a little stubborn with these kinds of questions in that i HAVE to understand them from every way...Bit silly really since i have the answer buts its reeeeally bugging me i can't get it another way.

4 Dice, what's the probability that at least two or more are the same number?

Well i've done it by considering the fact its 1 - P(they are all different numbers) leaving the probability that at least one must be the same. I started working it out a different way, which didn't work, so settled for the above.

Anywho, on my google travels, i came across this little table:

------------------------------------------Instances -- Probability
All four match ---------------------------- 6 ----------0.46%
two pairs of different numbers ---------90 --------- 6.94%
three dice match-------------------------120 -------- 9.26%
no matches------------------------------360 ---------27.78%
one pair matches ----------------------720 ------55.56%
TOTAL ---------------------------------1,296--------100.00%

Now, this 'one pair patches' is bugging me. How the hell do you get 720?

Because i was thinking:
4 dice.
Now if we choose any 2 of these dice (4 choose 2 = 6)
As there are 6 different ways to get pairs in two dice {(1,1), (2, 2)...etc}
There must be 6(number of ways to get 2 dice) x 6 instances.

What is it in my thinking that is incorrect?

Thankyou
• Jan 17th 2011, 10:22 PM
matheagle
I assume you want EXACTLY one pair.
I would look at it as 4 slots, each a toss.
Putting the match first there are 6 ways of having a pair in the first two positions.
(1,1).... (6,6) then for the last two tosses, there are 5 choose 2 numbers to select which is 10.
Now switching the positions, so that the pair need not land in the first two spots
there are 4!/(1!1!2!) ways of rewriting this sequence

\$\displaystyle 6{5\choose 2} 4!/2!=(6)(10)(12)=720\$
• Jan 17th 2011, 10:42 PM
Unknown008
Order is important here.

The ways to get exactly one pair is for example:
1 1 2 3
1 1 3 2
1 2 1 3
1 3 1 2
1 2 3 1
1 3 2 1
2 1 3 1
3 1 2 1
2 1 1 3
3 1 1 2
2 3 1 1
3 2 1 1

This is only to show the pairs you get with 1 and two other numbers 2 and 3

Now, there is for every single number 1 through 6!