A couple has four children. Assuming independence and that the probability of a boy is 1/2, determine the probability that the two oldest children are girls and the two youngest children are boys.
To determine the probability of the two oldest children being girls and the two youngest children being boys, we can break down the problem into separate events and calculate their probabilities individually.
First, let's consider the probability of each child being a girl or a boy. Since the probability of having a boy or a girl is assumed to be 1/2, we can label this as a binomial distribution with n = 1 (one child) and p = 1/2 (probability of success, which in this case is having a girl).
Now, let's calculate the probability for each event:
1. Probability of the oldest child being a girl = P(G1) = 1/2
2. Probability of the second oldest child being a girl = P(G2) = 1/2
3. Probability of the third youngest child being a boy = P(B3) = 1/2
4. Probability of the youngest child being a boy = P(B4) = 1/2
Since the events are independent, we can multiply the individual probabilities:
P(GG)BB = P(G1) * P(G2) * P(B3) * P(B4) = (1/2) * (1/2) * (1/2) * (1/2) = 1/16
Therefore, the probability that the two oldest children are girls and the two youngest children are boys is 1/16.