When throwing a fair die, what is the variance of the number of throws needed to get a 1?

To find the variance of the number of throws needed to get a 1 when throwing a fair die, we can model this scenario using a geometric distribution.

### Step 1: Identify the probability of success
Since the die is fair, the probability of getting a 1 on any given throw (success) is:
\[
p = \frac{1}{6}
\]

### Step 2: Variance formula for a geometric distribution
For a geometric distribution, the variance \( \text{Var}(X) \) is given by:
\[
\text{Var}(X) = \frac{1 - p}{p^2}
\]

### Step 3: Substitute the probability into the formula
Substitute \( p = \frac{1}{6} \) into the variance formula:
\[
\text{Var}(X) = \frac{1 - \frac{1}{6}}{\left(\frac{1}{6}\right)^2} = \frac{\frac{5}{6}}{\frac{1}{36}} = 5 \times 36 = 180
\]

### Final Answer
The variance of the number of throws needed to get a 1 is **180**.

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the expected number of throws needed to get a 1?
2. How would the variance change if the die were biased?
3. Can you derive the variance formula for a geometric distribution?
4. What is the probability of getting a 1 on the first throw?
5. How does the variance differ between a fair die and a loaded die?

**Tip:** In problems involving geometric distributions, always remember that the variance depends significantly on the probability of success. The lower the probability, the higher the variance.

Explore how to calculate the variance of the number of throws needed to get a 1 with a fair die. Understand the geometric distribution and its variance formula. Suitable for high school students studying probability theory.

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