Suppose that the New England Colonials baseball team is equally likely to win any particular game as not to win it. Suppose also that we choose a random sample of 30 Colonials games. (a)	Estimate the number of games in the sample that the Colonials win by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response.

Given that the New England Colonials baseball team is equally likely to win or not win any particular game, the probability of winning a game is \( p = 0.5 \). 

We are asked to find the mean (expected value) of the number of games won in a random sample of 30 games. 

### Step-by-Step Solution:

1. **Identify the type of distribution**:  
   The situation follows a binomial distribution because we are dealing with a fixed number of independent trials (30 games), each with two possible outcomes (win or not win).

   The binomial distribution is defined as:
   \[
   X \sim \text{Binomial}(n, p)
   \]
   where:
   - \( n = 30 \) (number of games)
   - \( p = 0.5 \) (probability of winning a game)

2. **Calculate the mean (expected value) of the distribution**:  
   The mean \( \mu \) of a binomial distribution is given by:
   \[
   \mu = n \times p
   \]
   Substituting the values we have:
   \[
   \mu = 30 \times 0.5 = 15
   \]

### Final Answer:
The mean number of games that the Colonials are expected to win in the sample of 30 games is \( \mathbf{15} \).

Would you like more details or have any further questions?

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Here are 8 related questions to consider:

1. What is the variance of the number of games the Colonials will win?
2. How would the expected number of wins change if the probability of winning was 0.6?
3. What is the probability that the Colonials win exactly 15 games out of the 30?
4. What is the standard deviation of the number of games won?
5. If the Colonials played 100 games, what would be the expected number of wins?
6. How would you calculate the probability of winning at least 20 games out of 30?
7. What is the probability of winning fewer than 10 games out of the 30?
8. How does the Central Limit Theorem apply to this binomial distribution?

**Tip**: For binomial distributions, knowing both the mean and variance is crucial for understanding the distribution's behavior.

Explore how to estimate the number of games won by the New England Colonials baseball team using a binomial distribution. Learn the formula for calculating the expected value in binomial distributions and its application to a random sample of 30 games. Suitable for advanced high school students.

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