Question

A coin is tossed three times. An outcome is represented by a string of the sort HTT (meaning a head on the first toss, followed by two tails). The 8 outcomes are listed in the table below. Note that each outcome has the same probability.

For each of the three events in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.

Outcomes Probability
HHH HTT THT HTH TTT TTH HHT THH
Event A: More tails than heads
Event B: Two or more tails
Event C: Alternating tail and head (with either coming first)

Answer 1

To analyze the events based on the outcomes of the coin tosses, let's first list the possible outcomes and then determine which outcomes belong to each event. After that, we will calculate the probability of each event.

### Outcomes
The 8 possible outcomes from tossing a coin three times are:
1. HHH
2. HTT
3. THT
4. HTH
5. TTT
6. TTH
7. HHT
8. THH

### Event A: More tails than heads
Outcomes with more tails than heads:
- TTT (3 tails)
- TTH (2 tails, 1 head)
- THT (2 tails, 1 head)
- HTT (2 tails, 1 head)

**Outcomes for Event A:** TTT, TTH, THT, HTT

### Event B: Two or more tails
Outcomes with two or more tails:
- TTT (3 tails)
- TTH (2 tails, 1 head)
- THT (2 tails, 1 head)
- HTT (2 tails, 1 head)

**Outcomes for Event B:** TTT, TTH, THT, HTT

### Event C: Alternating tail and head (with either coming first)
Outcomes that alternate between heads and tails:
- HTH (Head-Tail-Head)
- THT (Tail-Head-Tail)

**Outcomes for Event C:** HTH, THT

### Summary of Outcomes in Each Event
- **Event A:** TTT, TTH, THT, HTT
- **Event B:** TTT, TTH, THT, HTT
- **Event C:** HTH, THT

### Calculating the Probability of Each Event
Since there are 8 equally likely outcomes, the probability of each event can be computed as follows:

1. **Event A: 4 favorable outcomes (TTT, TTH, THT, HTT)**
\[
P(A) = \frac{4}{8} = \frac{1}{2}
\]

2. **Event B: 4 favorable outcomes (TTT, TTH, THT, HTT)**
\[
P(B) = \frac{4}{8} = \frac{1}{2}
\]

3. **Event C: 2 favorable outcomes (HTH, THT)**
\[
P(C) = \frac{2}{8} = \frac{1}{4}
\]

### Final Table
Now, let's summarize the results in a table:

| Outcomes | Probability |
|----------|-------------|
| HHH | |
| HTT | |
| THT | |
| HTH | |
| TTT | |
| TTH | |
| HHT | |
| THH | |
| Event A: More tails than heads | TTT, TTH, THT, HTT | 0.5 |
| Event B: Two or more tails | TTT, TTH, THT, HTT | 0.5 |
| Event C: Alternating tail and head (with either coming first) | HTH, THT | 0.25 |

This concludes the analysis of the events based on the outcomes of tossing a coin three times.