Question

An experiment consists of first rolling a die and then tossing a coin.

a. List the sample space.

b. Let A be the event that either a three or a four is rolled first, followed by landing a head on the coin toss. Find $P\left(A\right)$.

c. Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

Short answer

(a)

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

(b) $P\left(A\right)=\frac{1}{6}$

(c) The events A and B are mutually exclusive.

Step-by-step solution

Given information (part a)

An experiment consists of first rolling a die and then tossing a coin.

Explanation (part a)

Sample space:

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

Given information (part b)

An experiment consists of first rolling a die and then tossing a coin.

Explanation (part b)

Let A be that either a three or four is rolled first, followed by a head on the coin toss.

Total number of possible outcomes$=12$

Possible outcomes$=\left\{\right(3H),(4H\left)\right\}$

Total number of outcomes$=2$

Thus the probability that either a three or four is rolled first, followed by a head on the coin toss is calculated as

$$\begin{gathered} P\left(A\right)=\frac{\text{Total number of outcomes}}{\text{Total number of possible outcomes}} \\ P\left(A\right)=\frac{2}{12} \\ P\left(A\right)=\frac{1}{6} \end{gathered}$$

Given information (part c)

An experiment consists of first rolling a die and then tossing a coin.

Explanation (part c)

Let A be either a three or four is rolled first, followed by a head on the coin toss.

Possible outcomes$=\left\{\right(3H),(4H\left)\right\}$

Let B be the event that a number less than two is rolled first, followed by a head on the coin toss.

Possible outcomes$=\left\{\left(1H\right)\right\}$

We observe that there is no common outcome in events A and B. Thus, events A and B are mutually exclusive.