Question

Suppose you roll two fair 6-sided dice, and A is the event that both dice are even, and B is the event that the sum of the dice is 9 or more.

Hint:  2.4, and the very first problem of this worksheet quiz.

(a) Find P(A)

(b) Find P(B)

(c) Find P(A ∪ B)

(d) Find P(A^c ∩ B^c)

Answer 1

Sample space of rolling of 2 fair dice :

a)  Event A = Both dice are event = { (2,2) , (2,4) , (2,6) , (4,2) , (4,4) , (4,6) , (6,2) ,  (6,4) ,  (6,4) }

P(A) = 9/36 = 1/4

b)  Event B = Sum of dice is 9 or more = {(3,6) , (4,5) , (4,6) , (5,4) , (5,5), (5,6), (6,3), (6,4), (6,5), (6,6) }

P(B) = 10/36 = 5/18

c) (A ∩ B) = { (4,6) ,  (6,4) ,  (6,4) }

P(A ∩ B) = 3/36 = 1/12

We know

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

So, P(A ∪ B) = 9/36 + 10/36 - 3/36

= 16/36

= 4/9

d) Now event A^c = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4),

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

Event B^c = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5),, (4, 1), (4, 2), (4, 3), (4, 4), (5, 1), (5, 2), (5, 3), (6, 1), (6, 2) }

A^c ∩ B^c = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (3,5), (4, 1), (4, 3), (5, 1), (5, 2), (5, 3), (6, 1)}

So, P(A^c ∩ B^c ) = 20/36 = 5/9