Why is the answer 30/31 for at least one girl given at least one boy?

[tramp]

Hi
I have some difficulties with following question.

A family has five children. Assuming that the probability of a girl on each birth was 1/2 and that the five births were independent, what is the probability the family has at least one girl, given they have at least one boy?

My solution is 1-(1/2)^4 = 15/16

However according to the book correct answer it 30/31.
Any ideas?

 

[Stephen Tashi]

Hi
I have some difficulties with following question.

A family has five children. Assuming that the probability of a girl on each birth was 1/2 and that the five births were independent, what is the probability the family has at least one girl, given they have at least one boy?

My solution is 1-(1/2)^4 = 15/16

However according to the book correct answer it 30/31.
Any ideas?

Look at material on the binomial distribution.

The conditions in the problem are satisified if the family has any of the following combinations of genders:
(exactly 1 boy and 4 girls)
(exactly 2 boys and 3 girls)
(exactly 3 boys and 2 girls)
(exactly 4 boys and 1 girl)

The conditions are not satisfied by the combinations of genders:
(exactly 5 boys and 0 girls )
(exactly 0 boys and 5 girls )

The problem says we are "given" that the family has at least one boy, so you should look at the formula for conditional probability.

 

[tramp]

Thanks a lot Stephen.
I figured it out. Your post was a great help.