Question

The Medina-Zenith family's biggest life aspiration was to have a baby girl. Luck was not on their side. They had eight boys in a row.

They were determined to follow their aspiration and have the 9th child. But lo and behold, their ninth child is was a boy.
According to the family's doctor, it is very unlikely for a family to have eight boys in a row followed by a girl.
However, according to Professor C, if a couple has nine children, it is guaranteed that at least one of the children will be a girl. We want to know who is correct by answering the questions that follow.
A) Can Professor C guarantee that a family with nine children will have at least one girl? Explain or show how, with a mathematical or numerical argument.
(Hint:Use the Binomial distribution!!)
B) How many boys would you expect that a family of nine will have? Must Show ALL Calculations!!!

Answer 1

A) Professor C is correct in stating that a family with nine children is guaranteed to have at least one girl. This can be demonstrated using the concept of the binomial distribution.

The probability of having a boy or a girl in any given birth is equal, assuming a 50% chance for each. Therefore, the probability of having a girl in a single birth is 1/2 or 0.5.

To calculate the probability of having at least one girl in a family of nine children, we can use the complement rule. The complement of having at least one girl is having no girls, i.e., all boys.

P(At least one girl) = 1 - P(All boys)

Using the binomial distribution formula:

P(X = k) = (n C k) * p^k * (1 - p)^(n - k)

P(All boys) = P(X = 0) = (9 C 0) * (0.5^0) * (0.5^9)

Calculating this probability:

P(All boys) = (1) * (1) * (0.5^9) = 0.5^9 = 0.001953125

Therefore, P(At least one girl) = 1 - P(All boys) = 1 - 0.001953125 = 0.998046875

The probability of having at least one girl in a family of nine children is approximately 0.998 or 99.8%. Thus, Professor C is correct in stating that it is guaranteed to have at least one girl in a family of nine children.

B) To calculate the expected number of boys in a family of nine children, we can use the concept of the binomial distribution.

The expected value of a binomial distribution is given by the formula:

E(X) = n * p

In this case, n = 9 (total number of children) and p = 0.5 (probability of having a boy in any given birth).

Calculating the expected number of boys:

E(X) = 9 * 0.5 = 4.5

Therefore, we would expect a family of nine children to have approximately 4.5 boys. Note that since we cannot have a fractional number of boys, the expected value represents the average number of boys over a large number of families with nine children.