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
THT HHH TTT HTT TTH THH HHT HTH
Event A: A tail on both the first and the last tosses
Event B: A tail on the second toss
Event C: More heads than tails

Answer 1

Let's analyze the events and their corresponding outcomes in detail.

### Event A: A tail on both the first and the last tosses
The relevant outcomes for this event must have 'T' in both the first and last positions. The outcomes satisfying this condition are:
- THT
- TTT

So, the outcomes for Event A are: **THT, TTT**.

### Event B: A tail on the second toss
The relevant outcomes for this event must have 'T' in the second position. The outcomes satisfying this condition are:
- HTT
- TTH
- HHT
- TTT

So, the outcomes for Event B are: **HTT, TTH, HHT, TTT**.

### Event C: More heads than tails
Here we'll count the number of heads (H) and tails (T) in each outcome. An outcome has more heads than tails if it contains 2 or 3 heads. The outcomes satisfying this condition are:
- HHH (3 heads, 0 tails)
- HHT (2 heads, 1 tail)
- THH (2 heads, 1 tail)

So, the outcomes for Event C are: **HHH, HHT, THH**.

### Probability Calculation
There are a total of 8 outcomes when a coin is tossed three times. Since each outcome is equally likely, the probability for any specific outcome is \( \frac{1}{8} \).

#### For the events, let's calculate their probabilities:
- **Event A** has 2 favorable outcomes (THT, TTT):
\[
P(A) = \frac{2}{8} = \frac{1}{4}
\]

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

- **Event C** has 3 favorable outcomes (HHH, HHT, THH):
\[
P(C) = \frac{3}{8}
\]

### Summary of Outcomes and Probabilities
Now we can summarize the events, the corresponding outcomes, and their probabilities.

| Event | Outcomes | Probability |
|--------|-----------------------------|----------------|
| A | THT, TTT | \( \frac{1}{4} \) |
| B | HTT, TTH, HHT, TTT | \( \frac{1}{2} \) |
| C | HHH, HHT, THH | \( \frac{3}{8} \) |

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