An experiment consists of first rolling a die and then tossing a coin:
a. How many elements are there in the sample space?
b. Let A be the event that either a or 4 is rolled first, followed by landing a tail on the coin toss.
Present
c. Let be the event that an even number is rolled, followed by landing a tail on the coin toss. Are the
events A and B mutually exclusive?
No, they are not Mutually Exclusive
Yes, they are Mutually Exclusive
Hint: Mutually Exclusive

Ask by Delgado Gibson. in the United States

Mar 13,2025

Question

An experiment consists of first rolling a die and then tossing a coin:
a. How many elements are there in the sample space?
b. Let A be the event that either a or 4 is rolled first, followed by landing a tail on the coin toss.
Present
c. Let be the event that an even number is rolled, followed by landing a tail on the coin toss. Are the
events A and B mutually exclusive?
No, they are not Mutually Exclusive
Yes, they are Mutually Exclusive
Hint: Mutually Exclusive

Ask by Delgado Gibson. in the United States

Mar 13,2025

UpStudy AI · Accepted Answer

**a. Sample Space Size**

- The die has $$6$$ possible outcomes.
- The coin has $$2$$ possible outcomes.
- Thus, the number of elements in the sample space is  
  $$
  6 \times 2 = 12.
  $$

**b. Probability of Event $$A$$**

- Event $$A$$ is defined as rolling a $$1, 2, 3,$$ or $$4$$, followed by landing tails.
- The probability of rolling one of these $$4$$ outcomes is  
  $$
  \frac{4}{6} = \frac{2}{3}.
  $$
- The probability of obtaining tails on the coin toss is  
  $$
  \frac{1}{2}.
  $$
- Since these events are independent, the probability $$P(A)$$ is  
  $$
  \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}.
  $$

**c. Mutual Exclusivity of Events $$A$$ and $$B$$**

- Event $$A$$: Rolling a $$1, 2, 3,$$ or $$4$$ and tossing tails.
- Event $$B$$: Rolling an even number (i.e., $$2, 4,$$ or $$6$$) and tossing tails.
  
To check whether $$A$$ and $$B$$ are mutually exclusive, we look for any common outcomes.

- The outcomes in $$A$$ with tails are:  
  $$
  (1,\text{T}),\ (2,\text{T}),\ (3,\text{T}),\ (4,\text{T}).
  $$
- The outcomes in $$B$$ with tails are:  
  $$
  (2,\text{T}),\ (4,\text{T}),\ (6,\text{T}).
  $$
- The common outcomes between $$A$$ and $$B$$ are:  
  $$
  (2,\text{T}) \text{ and } (4,\text{T}).
  $$

Since there are common outcomes (i.e., the intersection is not empty), the events $$A$$ and $$B$$ are **not mutually exclusive**.
**a. Sample Space Size:** There are 12 elements in the sample space. **b. Probability of Event $$A$$:** $$ P(A) = \frac{1}{3} $$ **c. Mutual Exclusivity of Events $$A$$ and $$B$$:** No, events $$A$$ and $$B$$ are not mutually exclusive.

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