You roll the dice many times and are interested in the avera

You roll the dice many times and are interested in the average value of the product of the two numbers showing n_1 n_2. Calculate this product for all 36 possible different configurations of the two dice and take the average of these 36 values. How does your answer compare with that in example 2.87?

Solution

possible products are

36, 30 , 25 , 24, 20 , 18, 16, 15, 12, 10 , 9 , 8 , 6, 5 , 4 , 3 , 2 ,1

36 one way (6,6)

30 two ways (6,5) (5,6)

25 one way (5,5)

24 two ways (6,4)(4,6)

20 two ways(5,4) (4,5)

18 two ways ( 6,3) (3,6)

16 one way (4,4)

15 two ways (5,3) (3,5)

12 four ways( 4,3) (3,4) (6,2) (2,6)

10 2 ways (5,2) (2,5)

9 one way (3,3)

8 two ways (4,2) (2,4)

6 four ways (1,6) (6,1) (2,3) (3,2)

5 two ways (5,1) (1,5)

4 three ways (4,1) (1,4) (2,2)

3 two ways (3,1) (1,3)

2 two ways (1,2) (2,1)

1 one way (1,1)

Average of the product = 1/36 * ( 36 + 2*30 + 25 + 2*24 + 2*20 + 2*18 + 16 + 2*15 + 4*12 + 2*10 + 9 + 2*8 + 3*6 + 2*5 + 3*4 + 2*3 + 2*2 + 1*1 )

= 435 / 36 = 12.08