Chapter 4.2, Problem 34E

a.

To determine

List the outcomes in the sample space for the experiment.

Answer to Problem 34E

The outcomes in the sample space for the experiment are given by:

$\text{Sample space}=\left\{\begin{array}{l}\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right),\left(1,6\right),\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right),\\ \left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right),\left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right),\\ \left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\end{array}\right\}$

Explanation of Solution

Two fair die are rolled. First one is red and one is blue. The 36 outcomes in the sample space.

Sample space:

Set of all possible outcomes of probability experiment is called sample space of the experiment.

Here two fair dice are rolled. There were 36 equally likely outcomes. Outcomes are occurred in ordered pairs. First number and second number in each ordered pair represents number comes up in the red die and blue dies respectively. Each of the two numbers can take values 1 to 6.

Therefore,

$\text{Sample space}=\left\{\begin{array}{l}\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right),\left(1,6\right),\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right),\\ \left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right),\left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right),\\ \left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\end{array}\right\}$

b.

To determine

List the outcomes in the event A.

Answer to Problem 34E

The outcomes in the event A are given by:

$A=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),\left(5,5\right),\left(6,6\right)\right\}$

Explanation of Solution

It is given that event A denotes same number comes up on both dice.

Here two fair dice are rolled. Event A contains those ordered pairs with same number on both die.

Therefore,

$A=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),\left(5,5\right),\left(6,6\right)\right\}$

c.

To determine

List the outcomes in the event B.

Answer to Problem 34E

The outcomes in the event B are given by:

$B=\left\{\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\right\}$

Explanation of Solution

It is given that event B denotes red die comes up 6.

Here two fair dice are rolled. Event B contains those ordered pairs with first number is 6.

Therefore,

$B=\left\{\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\right\}$

d.

To determine

List the outcomes in the event C.

Answer to Problem 34E

The outcomes in the event C are given by:

$C=\left\{\left(1,6\right),\left(6,1\right)\right\}$

Explanation of Solution

It is given that event C denotes one die comes up 6 and the other die comes up 1.

Here two fair dice are rolled. There is only two possibilities for both dice 1 and 6 since C denotes one die comes up 6 and the other die comes up 1.

Therefore,

$C=\left\{\left(1,6\right),\left(6,1\right)\right\}$

e.

To determine

Check whether the events A and B are mutually exclusive or not. Explain the reason.

Answer to Problem 34E

Events A and B are not mutually exclusive.

Explanation of Solution

Mutually exclusive events:

Two or more events are said to be mutually exclusive if it is not possible to occur them together. That is, for mutually exclusive events there is no element in common.

Events A and B denotes that same number comes up on both die and red die comes up 6 respectively.

Here,

$\begin{array}{l}A=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),\left(5,5\right),\left(6,6\right)\right\}\\ B=\left\{\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\right\}\end{array}$

From the events, it is clear that outcome $\left(6,6\right)$ is common to both A and B. Therefore, events A and B are not mutually exclusive.

f.

To determine

Check whether the events A and C are mutually exclusive or not. Explain the reason.

Answer to Problem 34E

Events A and C are mutually exclusive.

Explanation of Solution

Events A and C denotes that same number comes up on both die and one die comes up 6 and the other die comes up 1 respectively.

Here,

$\begin{array}{l}A=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),\left(5,5\right),\left(6,6\right)\right\}\\ C=\left\{\left(1,6\right),\left(6,1\right)\right\}\end{array}$

From the events, it is clear that no outcome is common to both A and C. Therefore, events A and C are mutually exclusive.