Question

A husband and wife have three children. The outcome of the first child being a boy, the second a girl, and the third a girl can be represented by BGG. Determine each of the following:

a)sample space that describes all the orders of the possible genders of the children
b)the event that at least one child is a girl
c)the event that at least one child is boy
d)is the event in part (C) the complement of the event in part ((b)?

Answer 1

Conditions

A husband and wife have three children. The outcome of the first child being a boy, the second a girl, and the third a girl can be represented by BGG. Determine each of the following:

a) sample space that describes all the orders of the possible genders of the children

b) the event that at least one child is a girl

c) the event that at least one child is boy

d) is the event in part (C) the complement of the event in part ((b)?

Solution

a) The sample space is:

BBB,BBG,BGB,BGG,GBB,GGG,GBG

b) The event is at least 1 child is a girl is the opposite event to the event of 3 boys.

The probability of 3 boys will be is $\frac{1}{8}$, then the probability of at least 1 girl is $\frac{1 - 1}{8} = \frac{7}{8}$

c) The event of one child is a boy is the same ($\frac{1}{8}$), because the probability to have boy or girl in each of 3 moments is $\frac{1}{2}$.

d) The event of at least 1 girl and at least 1 boy is the opposite to event "all 3 boys or all 3 girls".

The probability of 3 boys is $\frac{1}{8}$, of 3 girls is $\frac{1}{8}$. Then of BBB or GGG is $\frac{1}{8} + \frac{1}{8} = \frac{1}{4}$.

And the probability of at least 1 girl and at least 1 boy is $\frac{1 - 1}{4} = \frac{3}{4}$.

Answer:

a) BBB,BBG,BGB,BGG,GBB,GGG,GBG

b) $\frac{1}{8}$

c) $\frac{1}{8}$

d) $\frac{3}{4}$