Let's analyze the task and answer it step-by-step.

Given Information

1. A coin is tossed three times, producing eight possible outcomes: $TTT, HHH, HTT, HTH, THH, TTH, THT,$ and $HHT$.

2. Each outcome has the same probability since each toss is independent with equal likelihood of landing heads or tails.

3. There are three events described with conditions on the outcomes:

   • Event A: Exactly one head.
   • Event B: A tail on the last toss.
   • Event C: Alternating head and tail (with either coming first).

Analysis of Each Event

1. Event A: Exactly one head

   • We need to identify outcomes with exactly one head.
   • Outcomes: $HTT, THT, TTH$
   • Probability: There are 3 favorable outcomes out of 8, so the probability is $\frac{3}{8}$.

2. Event B: A tail on the last toss

   • We need to identify outcomes where the last toss is a tail.
   • Outcomes: $TTT, HTT, HHT, THT$
   • Probability: There are 4 favorable outcomes out of 8, so the probability is $\frac{4}{8} = \frac{1}{2}$.

3. Event C: Alternating head and tail (with either coming first)

   • We need to identify outcomes that alternate head and tail.
   • Outcomes: $HTH, THT$
   • Probability: There are 2 favorable outcomes out of 8, so the probability is $\frac{2}{8} = \frac{1}{4}$.

Summary for the Table

• Event A (Exactly one head): Outcomes = $HTT, THT, TTH$, Probability = $\frac{3}{8}$
• Event B (A tail on the last toss): Outcomes = $TTT, HTT, HHT, THT$, Probability = $\frac{1}{2}$
• Event C (Alternating head and tail): Outcomes = $HTH, THT$, Probability = $\frac{1}{4}$

Would you like further clarification on any of these steps?

Related Questions

1. How would the probabilities change if the coin were biased?
2. What is the probability of getting at least two heads in three tosses?
3. How would you calculate the probability if the coin were tossed four times instead?
4. What are the possible outcomes if the coin is tossed twice?
5. How do you calculate the expected number of heads in three coin tosses?

Tip

For problems involving independent events with equal probability, listing all possible outcomes often simplifies identifying favorable outcomes for each event.