Question

Two fair dice are tossed together. Let X be the sum and Y the product of the two numbers on the top of the dice. Calculate E(X+ 3Y).

Answer 1

If two dice are tossed the possible outcomes are 1,2,3,4,5,6 on each die. To obtain sum and product.

Dice 1	Dice 2	X (sum )	Y (product)
1	1	2	1
1	2	3	2
1	3	4	3
1	4	5	4
1	5	6	5
1	6	7	6
2	1	3	2
2	2	4	4
2	3	5	6
2	4	6	8
2	5	7	10
2	6	8	12
3	1	4	3
3	2	5	6
3	3	6	9
3	4	7	12
3	5	8	15
3	6	9	18
4	1	5	4
4	2	6	8
4	3	7	12
4	4	8	16
4	5	9	20
4	6	10	24
5	1	6	5
5	2	7	10
5	3	8	15
5	4	9	20
5	5	10	25
5	6	11	30
6	1	7	6
6	2	8	12
6	3	9	18
6	4	10	24
6	5	11	30
6	6	12	36

table for X

X value (x)	Frequency (f)	x*f
2	1	2
3	2	6
4	3	12
5	4	20
6	5	30
7	6	42
8	5	40
9	4	36
10	3	30
11	2	22
12	1	12
Total	36	252

E(x) = sum(f*x)/sum(f) = 252/36 = 7

table for Y

Y value (y)	Frequency (f)	f*y
1	1	1
2	2	4
3	2	6
4	3	12
5	2	10
6	4	24
8	2	16
9	1	9
10	2	20
12	4	48
15	2	30
16	1	16
18	2	36
20	2	40
24	2	48
25	1	25
30	2	60
36	1	36
Total	36	441

E(y) = sum(f*y)/sum(f) = 441/36 = 12.25

E(X+ 3Y) = E(X) + 3*E(Y) = 7 + 12.25*3 = 7 + 36.75 = 43.75