Find the probability that in a family of 4 children there will be (a) at least 1 boy (b) at least 1 girl and 1 boy. Assume that the probability of a male birth is 1/2.

To find the probability of certain events in this scenario, we can use the concept of combinations along with the concept of probability.

a) To find the probability of having at least 1 boy in a family of 4 children, we need to find the probability of having 1 boy, 2 boys, 3 boys, and 4 boys. Since the probability of a male birth is 1/2, the probability of a female birth is also 1/2.

Let's calculate the probability of having exactly 1 boy:
- The number of ways to have exactly 1 boy in a family of 4 children is given by the combination formula: C(4, 1) = 4! / (1! * (4-1)!) = 4.
- The probability of having exactly 1 boy is given by the product of the number of ways and the probability of having a boy: 4 * (1/2)^1 * (1/2)^3 = 4/16 = 1/4.

Similarly, we can calculate the probabilities of having 2, 3, and 4 boys using the same approach. Then, sum up these probabilities to find the probability of at least 1 boy.

b) To find the probability of having at least 1 girl and 1 boy in a family of 4 children, we need to exclude the cases where there are only boys or only girls.

Again, let's calculate the probability of having exactly 1 girl and 1 boy:
- The number of ways to have exactly 1 girl and 1 boy in a family of 4 children is given by the combination formula: C(4, 1) * C(3, 1) = (4! / (1! * (4-1)!) * (3! / (1! * (3-1)!)) = 4 * 3 = 12.
- The probability of having exactly 1 girl and 1 boy is given by the product of the number of ways and the probability of each gender: 12 * (1/2)^2 * (1/2)^2 = 12/16 = 3/4.

Similarly, exclude the cases where there are only boys or only girls, and sum up the probabilities of the remaining cases to find the probability of at least 1 girl and 1 boy.

Please note that these calculations assume that the probability of a male birth is exactly 1/2 and that each child's gender is independent of the others.