Chapter 5.2, Problem 34E

Dice: Two fair dice are rolled. The first die is red and the second is blue. An experiment consists of observing the numbers that come up on the dice.

There are 36 outcomes in the sample space. They are ordered pairs of the form (Red die, Blue die). List the 36 outcomes.

Let A be the event that the same number comes up on both dice. List the outcomes A.

Let B be the event that the red die comes up 6. List the outcomes in B.

Let C be the event that one die comes up 6 and the other comes up 1. List the outcomes C.

Are events A and B mutually exclusive? Explain.

Are events A and C mutually exclusive? Explain.

To determine

To find: The list of $36$ outcomes as the ordered pairs of the form (Red die, Blue die).

Answer to Problem 34E

  $\{\begin{array}{l}(1,\text{ 1), }(1,\text{ 2), }(1,\text{ 3), }(1,\text{ 4), }(1,\text{ 5), }(1,\text{ 6), }(2,\text{ 1), }(2,\text{ 2), }(2,\text{ 3), }(2,\text{ 4), }(2,\text{ 5), }(2,\text{ 6),}\\ (3,\text{ 1), }(3,\text{ 2), }(3,\text{ 3), }(3,\text{ 4), }(3,\text{ 5), }(3,\text{ 6), }(4,\text{ 1), }(4,\text{ 2), }(4,\text{ 3), }(4,\text{ 4), }(4,\text{ 5), }(4,\text{ 6),}\\ (5,\text{ 1), }(5,\text{ 2), }(5,\text{ 3), }(5,\text{ 4), }(5,\text{ 5), }(5,\text{ 6), }(6,\text{ 1), }(6,\text{ 2), }(6,\text{ 3), }(6,\text{ 4), }(6,\text{ 5), }(6,\text{ 6)}\end{array}\}$

Explanation of Solution

Two fair dies are rolled. The first die is red and the second is blue. An experiment consists of observing the numbers that come up on the die.

If we toss a die there will be six outcomes $(1\text{, 2, 3, 4, 5, 6)}$ .

If two dies tossed together, when get “ $\text{1}$ ” in a die at the same time we might get $\text{"1" or "2" or "3" or "4" or "5" or "6"}$ in another die. Therefore altogether $36$ outcomes.

The possible outcomes as the ordered pairs of the form (Red die, Blue die) is

  $\{\begin{array}{l}(1,\text{ 1), }(1,\text{ 2), }(1,\text{ 3), }(1,\text{ 4), }(1,\text{ 5), }(1,\text{ 6), }(2,\text{ 1), }(2,\text{ 2), }(2,\text{ 3), }(2,\text{ 4), }(2,\text{ 5), }(2,\text{ 6),}\\ (3,\text{ 1), }(3,\text{ 2), }(3,\text{ 3), }(3,\text{ 4), }(3,\text{ 5), }(3,\text{ 6), }(4,\text{ 1), }(4,\text{ 2), }(4,\text{ 3), }(4,\text{ 4), }(4,\text{ 5), }(4,\text{ 6),}\\ (5,\text{ 1), }(5,\text{ 2), }(5,\text{ 3), }(5,\text{ 4), }(5,\text{ 5), }(5,\text{ 6), }(6,\text{ 1), }(6,\text{ 2), }(6,\text{ 3), }(6,\text{ 4), }(6,\text{ 5), }(6,\text{ 6)}\end{array}\}$

To determine

To find: The outcomes of the event $A$ , that the same number comes up in both dies.

Answer to Problem 34E

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

Explanation of Solution

In both dies, we have to get the same numbers. It means that, if we got “ $\text{1}$ ” in the red die than we need to get “ $\text{1}$ ” in blue die also.

Therefore the possible outcomes are

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

To determine

To find: The outcomes of the event $B$ , that the red die comes up $6$ .

Answer to Problem 34E

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

Explanation of Solution

In red die we need to get “ $6$ ” and in blue die, it may be any number.

Therefore the possible outcomes are

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

To determine

To find: The outcomes of the event $C$ , that one die comes up $6$ and the other comes up $1$ .

Answer to Problem 34E

  $C=\{(6,\text{ 1), }(1,6\text{)}\}$

Explanation of Solution

If $\text{6}$ comes up in red die then in blue die it should come up $1$ . Similarly, if $\text{6}$ comes up in blue die then in red die it should come up $1$ .

The possible outcome

  $C=\{(6,\text{ 1), }(1,6\text{)}\}$

To determine

To explain: Whether the two events $A\text{ and }B$ are mutually exclusive.

Answer to Problem 34E

No, both events have common elements.

Explanation of Solution

If two events are said to be mutually exclusive, it is impossible for both to occur. But the two events $A\text{ and }B$ havepossible occurrence ( $6,\text{ 6}$ ). Therefore it is not mutually exclusive.

To determine

To explain: Whether the two events $A\text{ and }C$ are mutually exclusive.

Answer to Problem 34E

Yes, both events have no any common outcomes.

Explanation of Solution

If the two events are said to be mutually exclusive, it is impossible for both to occur. Here, the two events $A\text{ and }C$ have no any possible occurrences. Therefore it is mutually exclusive.