Questions: Assume that boys and girls are equally likely. Find the probability that when a couple has three children, there are exactly 3 boys. What is the probability of exactly 3 boys out of three children? (Type an integer or a simplified fraction.)

$Assume that boys and girls are equally likely. Find the probability that when a couple has three children, there are exactly 3 boys. What is the probability of exactly 3 boys out of three children? (Type an integer or a simplified fraction.)$

Transcript text: Assume that boys and girls are equally likely. Find the probability that when a couple has three children, there are exactly 3 boys. What is the probability of exactly 3 boys out of three children? $\square$ (Type an integer or a simplified fraction.)

Solution

Solution Steps

Step 1: Identify the Specific Event of Interest

We are interested in the event where exactly 3 out of 3 children are boys (or girls).

Step 2: Calculate the Probability of the Specific Event

The probability of having exactly $k$ boys (or girls) out of $n$ children is given by the binomial probability formula: $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ Where $\binom33$ is the binomial coefficient representing the number of ways to choose 3 successes out of 3 trials, $p = 0.5$ is the probability of success (having a boy or girl) on each trial, and $1-p = 0.5$ is the probability of failure on each trial. Plugging in the values, we get: $$P(X = 3) = \binom{3}{3} \times 0.5^3 \times 0.5^{3-3}$$ $$P(X = 3) = 0.125$$