Question 1098975: two 6-sided fair dice are thrown n times. let x denote the number of throws in which the result on the first die exceeds the results on the second die. what is the distribution of x?
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website! Let's look at the throws when the second die is a 1.
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)
This would lead to x=5 out of 6.
Similarly for the second die equal to 2.
(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)
This gives x=4 out of 6.
All the way to the second die of 6.
(1,6) (2,6) (3,6) (4,6) (5,6) (6,6)
which has a x=0.
Summing them all,
So 15 out of 36 throws have the first dice exceeding the second.
Similarly 15 of 36 will have the second dice exceeding the first.
Finally 6 of 36 will have the first equal to the second.
There are no other possibilities.
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