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,

Question 1: Roll two fair dice. Then the sample space S is the
following.
S =
(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)
Let E be the event that the sum of the dice is odd, let F be the
event that the first die lands on 1,
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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
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