Question

Roll two dice simultaneously once. Let A be the event that the sum of the two dice is 8 and B be the event that at least one of the dice is odd.

a) Find P(A) and P(B). Express your answer as a FRACTION.

b) Find P(A given B) and P(B given A). Express your answer as FRACTIONS. Are A and B independent? Explain.

c) Find P(A and B). Express your answer as a FRACTION.

d) Find P(A or B). Express your answer as a FRACTION.

Answer 1

The sample space of rolling 2 dice are 36

n(S) = 36

A - the event that the sum of the two dice is 8

The outcomes for event A are

A = { (2,6),(3,5),(4,4),(5,3),(6,2)}

n(A) = 5

B - the event that at least one of the dice is odd

The outcomes for event B is

B = { (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)}

n(B) = 27

a) P(A) = n(A)/n(S) = 5/36

P(B) = n(B)/n(S) = 27/36 =3/4

b) P(A|B) = P(A and B)\P(B)

P(A and B) - The probability that the sum of the two dice is 8 and at least one of the dice is odd

The outcomes are {(3,5),(5,3)}

n(A and B) = 2

P(A and B) = n(A and B)/n(S) = 2/36

P(A|B) = 2/27

and P(B|A) = P(A and B)/ P(A) = 2/5

Are A and B independent?

For independet events P(A and B) = P(A)*P(B) = (5/36) * (27/36) = 5/48

But ,

Hence A and B are not independent.

c) P(A and B) = 2/36 = 1/18

d) P(A or B) = P(A) +P(B) -P(A and B) = (5+27-2) /36 = 30/36=5/6