an experiment consist of first rolling die and them tossing a coin

Question

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

(a) List the sample space.

1. {(3, H), (4, H)}
2. {(3, H), (3, T), (4, H), (4, T)}
3. {(1, T), (2, T), (3, T), (4, T), (5, T), (6, T)}
4. {(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)}
5. {(1, H), (2, H), (3, H), (4, H), (5, H), (6, H)}

(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(A). (Enter your probability as a fraction.)

P(A) = ______

(c) Suppose that a new experiment consists of first rolling a die and then tossing a coin twice. Let B be the event that the first and second coin tosses land on heads. Let C be the event that either a three or a four is rolled first, followed by landing a head on the first coin toss. Are the events B and C mutually exclusive? Explain your answer.

1. Events B and C are mutually exclusive because they have different probabilities.
2. Events B and C are mutually exclusive because the first and second coin tosses cannot land on heads when a three or four is rolled first.
3. Events B and C are not mutually exclusive because the first and second coin tosses can land on heads when a three or four is rolled first.
4. Events B and C are mutually exclusive because they are dependent events.

Answer 1

S₁ = {(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)}

A = {(3, H), (4, H)}

P(A) = 2/12 = 1/6

The probability of rolling either a 3 or 4 on a die and then getting a "Head" is 1/6.

S₂ = {(1, HH), (1, HT), (1, TH), (1, TT), (2, HH), (2, HT), (2, TH), (2, TT), (3, HH), (3, HT), (3, TH), (3, TT), (4, HH), (4, HT), (4, TH), (4, TT), (5, HH), (5, HT), (5, TH), (5, TT), (6, HH), (6, HT) (6, TH), (6, TT)}

B = {(1, HH), (2, HH), (3, HH), (4, HH), (5, HH), (6, HH)}

C = {(3, HH), (3, HT), (4, HH), (4, HT)}

P(B ∩ C) = 2/24 = 1/12

Events B and C are not mutually exclusive because the first and second coin tosses can land on heads when a three or four is rolled first. The events are overlapping.

Answer 2

Probability rolling a 3 or a 4 = 2/6 = 1/3

Probability of getting a head = 1/2

Probability of both occurring = 1/3 * 1/2 = 1/6 (since they are independent events)

Events B and C are not mutually exclusive because the first and second coin tosses can land on heads when a three or four is rolled first.

Answer 3