(b)Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

To quantify the uncertainty of the estimate, we need to calculate the standard deviation of the binomial distribution.

### Step-by-Step Solution:

1. **Identify the variance of the binomial distribution**:  
   The variance \( \sigma^2 \) of a binomial distribution is given by:
   \[
   \sigma^2 = n \times p \times (1 - p)
   \]
   where:
   - \( n = 30 \) (number of games)
   - \( p = 0.5 \) (probability of winning a game)

   Substituting the values:
   \[
   \sigma^2 = 30 \times 0.5 \times (1 - 0.5) = 30 \times 0.5 \times 0.5 = 7.5
   \]

2. **Calculate the standard deviation**:  
   The standard deviation \( \sigma \) is the square root of the variance:
   \[
   \sigma = \sqrt{7.5} \approx 2.7386
   \]

### Final Answer:
The standard deviation of the distribution, rounded to at least three decimal places, is \( \mathbf{2.739} \).

Would you like more details or have any further questions?

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Here are 8 related questions to consider:

1. How does the standard deviation change if the probability of winning is 0.7 instead of 0.5?
2. What is the probability of winning between 12 and 18 games inclusive?
3. How does the standard deviation affect the shape of the binomial distribution?
4. If the sample size was increased to 60 games, what would be the new standard deviation?
5. Can you interpret what the standard deviation tells us about the Colonials' performance?
6. What would the variance be if the probability of winning was 0.4?
7. How would you calculate the probability of winning exactly 20 games out of 30 using the standard deviation?
8. How does the standard deviation compare to the expected value in terms of understanding the distribution?

**Tip**: Understanding the standard deviation is key to grasping how spread out the possible outcomes are around the mean in a distribution.

Learn how to quantify uncertainty in estimates using standard deviation in the binomial distribution. This problem involves calculating variance and standard deviation based on a binomial distribution with specific parameters. Suitable for advanced high school students and anyone interested in probability theory.

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