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.



Let's first list the eight possible outcomes when tossing a coin three times. Since each outcome is a combination of heads (H) and tails (T), they can be represented as follows:

1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT

Now, let's define the three events that we'll be analyzing:

- **Event A**: The first toss is a head (H).
- **Event B**: Exactly two heads occur.
- **Event C**: At least two tails occur.

### Step 1: List outcomes contained in each event
We will now list the outcomes for each event:

- **Event A (first toss is H)**: The outcomes where the first toss is H are:
  - HHH
  - HHT
  - HTH
  - HTT

- **Event B (exactly two heads)**: The outcomes with exactly two heads are:
  - HHT
  - HTH
  - THH

- **Event C (at least two tails)**: The outcomes with at least two tails are:
  - HTT
  - THT
  - TTH
  - TTT

### Step 2: Calculate the probabilities
Since there are 8 possible outcomes, and each outcome is equally likely, the probability of any event is the number of favorable outcomes divided by 8.

- **Event A** has 4 favorable outcomes: \( \frac{4}{8} = 0.5 \).
- **Event B** has 3 favorable outcomes: \( \frac{3}{8} = 0.375 \).
- **Event C** has 4 favorable outcomes: \( \frac{4}{8} = 0.5 \).

### Final Table
| Outcome | Event A (First H) | Event B (Two Heads) | Event C (At Least Two Tails) |
|---------|-------------------|---------------------|-----------------------------|
| HHH     | ✔                 | ✔                   | ✘                           |
| HHT     | ✔                 | ✔                   | ✘                           |
| HTH     | ✔                 | ✔                   | ✘                           |
| HTT     | ✔                 | ✘                   | ✔                           |
| THH     | ✘                 | ✔                   | ✘                           |
| THT     | ✘                 | ✘                   | ✔                           |
| TTH     | ✘                 | ✘                   | ✔                           |
| TTT     | ✘                 | ✘                   | ✔                           |

Probabilities:
- **P(A) = 0.5**
- **P(B) = 0.375**
- **P(C) = 0.5**

Let me know if you'd like more details or clarifications!

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Here are 5 related questions:

1. What is the probability of getting no heads at all in three tosses?
2. What is the probability of getting exactly one tail in three tosses?
3. How would the probabilities change if the coin was biased, favoring heads 60% of the time?
4. How do we calculate the expected number of heads in three tosses?
5. What is the probability of getting at least one head in three tosses?

**Tip**: When outcomes are equally likely, the probability of an event is simply the ratio of favorable outcomes to the total possible outcomes.

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.

Explore how to calculate the probability of different outcomes when tossing a coin three times. Learn about key concepts such as the probability formula and combinatorics, suitable for students in Grades 6-8.

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