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Chapter 9: Problem 510

In a family of 4 children, what is the probability that there will be exactly two boys?

Short Answer

Expert verified
The probability that there will be exactly 2 boys in a family of 4 children is \(\frac{3}{8}\).

Step by step solution

01

Identify and list down the known values

We are given the following information: - n = 4 (There are 4 children) - k = 2 (We want exactly 2 boys) - p = probability of a boy = 1/2 (Assuming equal probability for both genders)
02

Compute the number of combinations (C(4, 2))

We need to find the number of ways to choose 2 boys (success) from 4 children (trials). We will use the formula for combinations: \[C(n, k) = \frac{n!}{k!(n - k)!}\] Substitute the values of n and k: \[C(4, 2) = \frac{4!}{2!(4 - 2)!} = \frac{4 \times 3 \times 2}{(2 \times 1)(2 \times 1)} = 6\] So, there are 6 ways to choose 2 boys from 4 children.
03

Calculate the probability of exactly 2 boys using the binomial probability formula

Now, we can use the binomial probability formula to find the probability of exactly 2 boys: \[P(K = 2) = C(n, k) \times p^k \times (1 - p)^{(n - k)}\] Substitute the values we found earlier: \[P(K = 2) = 6 \times \left(\frac{1}{2}\right)^2 \times \left(\frac{1}{2}\right)^{(4 - 2)}\] \[P(K = 2) = 6 \times \frac{1}{4} \times \frac{1}{4} = \frac{3}{8}\] So, the probability that there will be exactly 2 boys in a family of 4 children is 3/8.

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