Question

Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is even. Are A,B independent events? Please show work where applicable.

Answer 1

Given that, Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is even.

i) There are, 3 outcomes that the red die comes up even, that are, { 2, 4, 6}

Thefore the probability of event A is, P(A) = 3/6 = 1/2 = 0.5

ii) There are total 36 outcomes when we rolled two dice and there are total 18 outcomes that the sum of the results is an even number and these outcumes are as follows,

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

Therefore, the probability that the sum of the results is an even number = 18/36 = 1/2

That is, P(event B) = 1/2 = 0.5

iii) Now, we find P(A and B)

There are 9 outcomes which gives even number on red dice and sum of the two dices ( red and greeen) is even number.

{ (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6) }

Therefore, P(A and B) = 9/36 = 1/4 = 0.25

Two events A and B are independent if,

P(A and B) = P(A) * P(B)

here, P(A) * P(B) = 1/2 * 1/2 = 1/4 = 0.25

Hence, P(A and B) = P(A) * P(B) = 1/4 = 0.25

Event A and B are independent.