Question

Roll two dice (one red and one white). Denote their outcomes as X1 and X2. Let T = X1+X2 denote the total, let X1 W X2 denote the maximum and let X1 V X2 denote the minimum. Find the following probabilities: (a) P(X1 ≥ 3|X2 ≤ 4) (b) P(T is prime) (c) P(T ≤ 8|X1 W X2 = 5) (d) P(X1 V X2 ≤ 5|T ≥ 8) (e) P(X1 W X2 ≥ 3|X1 W X2 ≤ 3)

Answer 1

[(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)

(1,2), (2,2), (3,2), (4,2), (5,2), (6,2)

(1,3), (2,3), (3,3), (4,3), (5,3), (6,3)

(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)

(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)

(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)]

Where left input is X1 and right one is X2

a) X1 is greater than equal to 3 in 24 cases

X2 is less than four in 24 cases

b) T is prime when either both X1 and X2 are even or odd

X1\ge 3| X2\le 4By above table we can see it happens exactly 18 times

c) T is less than equal to 8, 26 times

Maximum of X1 and X2 is equal to 5, 9 times

out of those 9 times T is less than equal to 8 only 4 times

d) Minimum of X1 and X2 is less than equal to 5, 25 times

T is greater than equal to 8, 15 times

out of these 15 times minimum of X1 and X2 is less than equal to 5, 6 times