Question

1. Two dice are rolled. There are 36 possible outcomes, the sample space is:

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

A = ‘second roll is a 6’

B = ‘sum of two dice equals 7’

C = ‘sum of two dice equals 3’

a. What is P(B|A)?

b. What is P(B|C)?

c. Are A and B independent? What about A and C? Show why or why not.

d. Are A and B mutually exclusive? What about A and C? Show why or why not.

e. What is P(A or B’)?

Answer 1

In the solution, for any event E, we will use the notation as the number of possible outcomes favourable to E. We will also denote the sample space by  .

Part (a)

Part (b)

Part (c)

Here,

Again,

and,

Hence, and thus A and B are independent events.

Again,

Also,

Thus, and hence A and C are not independent.

Part (d)

We have already shown in Part (c) that and .

Hence, A and C are mutually exclusive events, while A and b are not.

Part (e)

Assuming that means , i.e. complement of B.

Now,

So,