Question

Suppose two dice are rolled. Find the probabilities of the following events.

a) the maximum of the two numbers rolled is less than or equal to 2;

b) the maximum of the two numbers rolled is less than or equal to 3;

c) the maximum of the two numbers rolled is exactly equal to 3.

d) Repeatb)andc)forxinsteadcf3,foreachxfrom1to6.

e) Denote by P(x) the probability that the maximum number is exactly x. What should P(I) +P(2) +P(3) +P(4) +P(5) +P(6) equal? Check this for your answers to d).

Answer 1

a)

X be highest of two values
outcome	X
(1,1)	1
(1,2)(2,1)(2,2)	2

required probability = 4/36 = 1/9

b)

X be highest of two values
outcome	X
(1,1)	1
(1,2)(2,1)(2,2)	2
(1,3)(3,1)(2,3)(3,2)(3,3)	3

total outcome=9

required probability =9/36 = 1/4

c)

outcome	X
(1,3)(3,1)(2,3)(3,2)(3,3)	3

required probability = 5/36

d)

for b)

let two numbers rolled is less than or equal to X

X	P(X)
≤1	1/36
≤2	4/36
≤3	9/36
≤4	16/36
≤5	25/36
≤6	36/36

for c)

X be highest of two values
outcome	X	P(X)
(1,1)	1	1/36
(1,2)(2,1)(2,2)	2	3/36
(1,3)(3,1)(2,3)(3,2)(3,3)	3	5/36
(1,4)(4,1)(2,4)(4,2)(3,4)(4,3)(4,4)	4	7/36
(1,5)(5,1)(2,5)(5,2)(3,5)(5,3)(4,5)(5,4)(5,5)	5	9/36
(1,6)(6,1)(2,6)(6,2)(3,6)(6,3)(4,6)(6,4)(5,6)(6,5)(6,6)	6	11/36

e)

X	P(X)
1	1/36
2	3/36
3	5/36
4	7/36
5	9/36
6	11/36

P(I) +P(2) +P(3) +P(4) +P(5) +P(6) = 36/36 = 1