Question

Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple having 5 children, what is the probability that they have at least 3 boys?

1) 0.3125
2) 0.375
3) 0.4375
4) 0.5

Answer 1

To determine the probability that a couple with 5 children will have at least 3 boys, we calculate the binomial probabilities for having exactly 3, 4, and 5 boys, and then sum these probabilities.

Step-by-step explanation:

The probability that a couple having 5 children will have at least 3 boys can be calculated using the binomial probability formula. The probability (P) of having exactly k boys out of n children, where the probability of having a boy is 0.5 (thus, p = 0.5 and q = 1 - p = 0.5 since it's equally likely to have a boy or a girl), is given by:

P(X = k) = C(n, k) * p^k * q^(n - k)

Where C(n, k) is the number of combinations of n items taken k at a time. Therefore, we'll calculate the probabilities of having 3, 4, and 5 boys separately and then add them together to get the total probability:

• P(X = 3) = C(5, 3) * (0.5)^3 * (0.5)^(5-3)
• P(X = 4) = C(5, 4) * (0.5)^4 * (0.5)^(5-4)
• P(X = 5) = C(5, 5) * (0.5)^5 * (0.5)^(5-5)

After calculating the values for P(X = 3), P(X = 4), and P(X = 5), summation of these values gives us the total probability of having at least 3 boys.

Adding the probabilities together, we find the total probability (which should match one of the options provided):

P(at least 3 boys) = P(X = 3) + P(X = 4) + P(X = 5)