Question

While it is often assumed that the probabilities of having a boy or a girl are the same, the actual probability of having a boy is slightly higher at 0.51. Suppose a couple plans to have 3 kids. What is the probability that they will have at least 2 boys?

1) 0.264
2) 0.382
3) 0.488
4) 0.624

Answer 1

To find the probability of a couple having at least 2 boys out of 3 kids, we calculate the probability of having exactly 2 boys and exactly 3 boys and add them together. The probability of having exactly 2 boys is calculated using the binomial probability formula, resulting in a probability of approximately 0.378135. The probability of having exactly 3 boys is simply 0.51 raised to the power of 3, which is 0.133651. Adding these probabilities together gives a probability of approximately 0.511786, which is closest to option 4) 0.624.

Step-by-step explanation:

To find the probability that a couple will have at least 2 boys out of 3 kids, we can calculate the probability of having exactly 2 boys and the probability of having exactly 3 boys and add them together.

To calculate the probability of having exactly 2 boys out of 3 kids, we use the binomial probability formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. In this case, the number of trials is 3, the number of successes is 2, and the probability of success (having a boy) is 0.51.

Using the formula, we get P(X=2) = C(3,2) * 0.51^2 * (1-0.51)^(3-2) = 3 * 0.51^2 * (0.49)^1 = 3 * 0.2601 * 0.49 = 0.378135.

To calculate the probability of having exactly 3 boys out of 3 kids, we simply raise the probability of having a boy (0.51) to the power of 3: P(X=3) = 0.51^3 = 0.133651.

Finally, we add the two probabilities together: P(at least 2 boys) = P(X=2) + P(X=3) = 0.378135 + 0.133651 = 0.511786.

Therefore, the probability that the couple will have at least 2 boys out of 3 kids is approximately 0.511786, which is closest to option 4) 0.624.