Question

# 1. Two dice are rolled. There are 36 possible outcomes, the sample space is: (1,1) (1,2)...

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’)?

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,

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