Binomial Probability Distribution.

Assume that each child has probability of 0.5 of being a boy. Find the probability that a family of five children has exactly one boy.

To solve this problem using the Binomial Probability Distribution, we'll use the formula for the probability of getting exactly \( k \) successes (boys) in \( n \) independent trials (children), where each trial has a success probability \( p \).

The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]

Here, \( n = 5 \), \( k = 1 \), and \( p = 0.5 \).

1. **Calculate the binomial coefficient** \(\binom{n}{k}\):
\[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]

\[ \binom{5}{1} = \frac{5!}{1!(5 - 1)!} = \frac{5!}{1! \cdot 4!} = \frac{5 \cdot 4!}{1 \cdot 4!} = 5 \]

2. **Calculate the probability** \( p^k \):
\[ p^k = 0.5^1 = 0.5 \]

3. **Calculate the probability** \( (1 - p)^{n - k} \):
\[ (1 - p)^{n - k} = (1 - 0.5)^{5 - 1} = 0.5^4 = \frac{1}{16} \]

4. **Combine these values to find the probability**:
\[ P(X = 1) = \binom{5}{1} \cdot 0.5^1 \cdot 0.5^4 = 5 \cdot 0.5 \cdot \frac{1}{16} \]

\[ P(X = 1) = 5 \cdot \frac{0.5}{16} = 5 \cdot \frac{1}{32} = \frac{5}{32} \]

So, the probability that a family of five children has exactly one boy is \(\frac{5}{32}\), which is approximately 0.15625 or 15.625%.

Would you like more details on this calculation or have any other questions?

Here are 8 related questions you might find interesting:

1. What is the probability that a family of five children has exactly two boys?
2. How do you calculate the cumulative probability for a binomial distribution?
3. What is the expected number of boys in a family of five children?
4. What is the variance and standard deviation of the number of boys in a family of five children?
5. How does the probability change if the probability of having a boy is 0.6 instead of 0.5?
6. Can you derive the binomial distribution formula?
7. How would you calculate the probability of having at least one boy in a family of five children?
8. What is the probability that all five children are boys?

**Tip:** When dealing with binomial distributions, it's important to understand the parameters \( n \) (number of trials) and \( p \) (probability of success), as they define the shape and characteristics of the distribution.

Learn how to calculate the probability of exactly one boy in a family of five children using the binomial probability formula. Step-by-step solution provided with calculations. Suitable for Grades 10-12.

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