Thread: Dice Pair Matching

4 players roll a die. They score 1 point if there is a matching pair. What are the probabilities of getting the possible scores.

For example: for the case where 2 people get ”1”, and the other two roll a ”3” and a ”4” respectively, then the team’s score would be 1. Similarly, for the case that 3 people throw a ”5” and the other one does not, the score would be 3, since there are 3 different pairs of players that have the same number.

Proposed solution:
Imagine players as slots to fill with possibilities of rolling dice.

Score 0:
All 4 players roll different numbers. So 6*5*4*3 = 360 ways of getting score 0.

Score 1:
2 Players roll the same, other 2 are different from each other and the matching pair. So 6*1*5*4 = 120 ways of getting score 0.

Score 2:
2 pairs of players match. So 6*1*5*1 = 30 ways. The ones are the corresponding matches to the preceding rolls.

Score 3:
3 matches and 1 different so 6*1*1*5 = 30 ways again. The 4th person can have any one of the 5 choices left over.

Score 4:
All roll the same number so. 6*1*1*1 = 6 ways of getting score 4.

My problem is these all add up to 546, whereas I would imagine they'd have to add up to possibilities.

Any suggestions?

Let’s look at the case score 2. That could happen if the players roll .
But that can happen is ways.
Moreover, there are ways to have two different pairs.
It seems to me that have under-counted this case.

If I have miss-read the problem please say why.

Hello, utopiaNow!

You're not considering WHO gets the pairs, etc.

Four players roll a die.
They score 1 point if there is a matching pair.
What are the probabilities of getting the possible scores?

For example:
2 people get ”1”, and the other two roll a ”3” and a ”4” resp., then the score would be 1.
Similarly, that 3 people throw a ”5” and the other one does not, the score would be 3.

Score 0
Four different numbers:
Number of ways: .


Score 1
A pair and two singles:
There are: . distributions . . .
Number of ways: .


Score 2
Two pairs:
There are 3 distributions.
. . If the four players are , they can be paired in 3 ways:
. . . .
Number of ways: .


Score 3
A triple and a single:
There are 4 choices of who gets the triple.
Number of ways: .


Score 4
A quadruple:
The only choice is the value of the quadruple.
Number of ways: .


Check: .

Originally Posted by Soroban

Hello, utopiaNow!

You're not considering WHO gets the pairs, etc.


Score 0
Four different numbers:
Number of ways: .


Score 1
A pair and two singles:
There are: . distributions . . .
Number of ways: .


Score 2
Two pairs:
There are 3 distributions.
. . If the four players are , they can be paired in 3 ways:
. . . .
Number of ways: .


Score 3
A triple and a single:
There are 4 choices of who gets the triple.
Number of ways: .


Score 4
A quadruple:
The only choice is the value of the quadruple.
Number of ways: .


Check: .

Ah, yes that's it. Thanks Soroban for your help and explanation.