Construct a probability distribution of getting heads in a three-coin toss?

To construct the probability distribution of getting heads in a three-coin toss, we need to list all possible outcomes, count the number of heads in each outcome, and then find the probabilities for each possible number of heads.

### Step 1: List all possible outcomes.
When tossing three coins, each coin can either be heads (H) or tails (T). The total number of possible outcomes is \(2^3 = 8\). Here are the outcomes:
- HHH (3 heads)
- HHT (2 heads)
- HTH (2 heads)
- HTT (1 head)
- THH (2 heads)
- THT (1 head)
- TTH (1 head)
- TTT (0 heads)

### Step 2: Count the number of heads for each outcome.
- 3 heads: HHH
- 2 heads: HHT, HTH, THH
- 1 head: HTT, THT, TTH
- 0 heads: TTT

### Step 3: Calculate the probabilities.
The probability of each individual outcome is \( \frac{1}{8} \), since all outcomes are equally likely. Now, we group the outcomes by the number of heads and calculate the probability for each:

- **0 heads**: 1 outcome (TTT) → Probability \( P(X = 0) = \frac{1}{8} \)
- **1 head**: 3 outcomes (HTT, THT, TTH) → Probability \( P(X = 1) = \frac{3}{8} \)
- **2 heads**: 3 outcomes (HHT, HTH, THH) → Probability \( P(X = 2) = \frac{3}{8} \)
- **3 heads**: 1 outcome (HHH) → Probability \( P(X = 3) = \frac{1}{8} \)

### Step 4: Probability Distribution Table

| Number of Heads (X) | Probability \( P(X) \) |
|---------------------|------------------------|
| 0                   | \( \frac{1}{8} \)      |
| 1                   | \( \frac{3}{8} \)      |
| 2                   | \( \frac{3}{8} \)      |
| 3                   | \( \frac{1}{8} \)      |

This table represents the probability distribution of getting heads in a three-coin toss.

Would you like further details or have any questions on this?

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Here are 5 related questions to expand on this:
1. What is the expected number of heads in a three-coin toss?
2. How would the distribution change for a four-coin toss?
3. What is the probability of getting at least two heads?
4. How can this distribution be modeled using a binomial distribution?
5. What is the cumulative probability of getting fewer than 2 heads?

**Tip:** A binomial distribution can be used for any number of coin tosses, with heads or tails representing a success or failure in each trial.

Explore the probability distribution of getting heads in a three-coin toss. This detailed solution breaks down all possible outcomes and calculates the probabilities for 0, 1, 2, and 3 heads using key probability concepts and the binomial distribution formula. Ideal for students in Grades 7-10.

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