Question

Suppose that you roll a die and your score is the number shown on the die. On the other
hand, suppose that your friend rolls five dice and his score is the number of 6’s shown out of five rollings. Compute the probability
(a) that the two scores are equal.
(b) that your friend’s score is strictly smaller than yours.

Answer 1

Please rate. Cheers !!!

X1 and X2 are the random variables representing the score of me (I) and my friend (II) respectively.

I have used I notation for me and II for my friend.

Below is the table -

me		Friend
Score	P(X1)	Score	P (X2)	Combination	P (X1) * P(X2)
1	0.166667	0	0.401878	P1	0.066979667
2	0.166667	0	0.401878	P2	0.066979667
2	0.166667	1	0.401878	P3	0.066979667
3	0.166667	0	0.401878	P4	0.066979667
3	0.166667	1	0.401878	P5	0.066979667
3	0.166667	2	0.160751	P6	0.026791833
4	0.166667	0	0.401878	P7	0.066979667
4	0.166667	1	0.401878	P8	0.066979667
4	0.166667	2	0.160751	P9	0.026791833
4	0.166667	3	0.03215	P10	0.005358333
5	0.166667	0	0.401878	P11	0.066979667
5	0.166667	1	0.401878	P12	0.066979667
5	0.166667	2	0.160751	P13	0.026791833
5	0.166667	3	0.03215	P14	0.005358333
5	0.166667	4	0.003215	P15	0.000535833
6	0.166667	0	0.401878	P16	0.066979667
6	0.166667	1	0.401878	P17	0.066979667
6	0.166667	2	0.160751	P18	0.026791833
6	0.166667	3	0.03215	P19	0.005358333
6	0.166667	4	0.003215	P20	0.000535833
6	0.166667	5	0.000129	P21	0.0000215
				Sum	0.86111183