# Family of Three Children - Conditional Probability

#### joshuaa

From families with three children, a family is selected at random and found to have
a boy. What is the probability that the boy has (a) an older brother and a younger
sister; (b) an older brother; (c) a brother and a sister? Assume that in a three-child
family all gender distributions have equal probabilities.

#### Plato

MHF Helper
From families with three children, a family is selected at random and found to have
a boy. What is the probability that the boy has (a) an older brother and a younger
sister; (b) an older brother; (c) a brother and a sister? Assume that in a three-child
family all gender distributions have equal probabilities.
Here is the probability space: $\begin{array}{*{20}{c}}B&B&B\\B&B&G\\B&G&B\\B&G&G\\G&B&B\\G&B&G\\G&G&B\\G&G&G\end{array}$ You are given that the family has a boy.
That creates a subspace. Now we just count.
a) is $\dfrac{1}{7}$. Please explain that.

#### joshuaa

First, the book answer for (a) is 1/14

-------------------------
But let me try to tell you why you got 1/7

we have 8 outcomes in the sample space
the probability of selecting a boy is 7/8 since only G G G does not have a boy

B B G and B G B and G B B has two boys and a girl
so the probability of getting two boys and a girl is 3/8

but if order is important only B B G will be the outcome of a boy has an older brother and a sister
so the probability is 1/8

Therefore, the answer for (a) is (1/8) / (7/8) = 1/7

Is the answer of the book wrong?

#### Plato

MHF Helper
First, the book answer for (a) is 1/14
we have 8 outcomes in the sample space
the probability of selecting a boy is 7/8 since only G G G does not have a boy
B B G and B G B and G B B has two boys and a girl
so the probability of getting two boys and a girl is 3/8
but if order is important only B B G will be the outcome of a boy has an older brother and a sister
so the probability is 1/8
Therefore, the answer for (a) is (1/8) / (7/8) = 1/7
Is the answer of the book wrong?
If someone tell you that a family has three children and at least one is a boy, what is the probability of BBG? clearly order does matter.
What is the probability of two boys and a girl? (order does not matter)
Yes, I argue the answer is 1/7 as written. I would like to hear what that author says.

#### romsek

MHF Helper
First, the book answer for (a) is 1/14

-------------------------
But let me try to tell you why you got 1/7

we have 8 outcomes in the sample space
the probability of selecting a boy is 7/8 since only G G G does not have a boy

B B G and B G B and G B B has two boys and a girl
so the probability of getting two boys and a girl is 3/8

but if order is important only B B G will be the outcome of a boy has an older brother and a sister
so the probability is 1/8

Therefore, the answer for (a) is (1/8) / (7/8) = 1/7

Is the answer of the book wrong?
I think it's $\dfrac{1}{14}$ because getting GBB could be either the "boy" having an older brother as we want, or
it could be that the "boy" is the older brother and thus has a younger brother.

Both are equiprobable and they total to $\dfrac{1}{7}$ in probability so they are both $\dfrac{1}{14}$ in probability.

#### Plato

MHF Helper
From families with three children, a family is selected at random and found to have
a boy. What is the probability that the boy has (a) an older brother and a younger
sister; (b) an older brother; (c) a brother and a sister? Assume that in a three-child
family all gender distributions have equal probabilities.

I think it's $\dfrac{1}{14}$ because getting GBB could be either the "boy" having an older brother as we want, or
it could be that the "boy" is the older brother and thus has a younger brother.
Both are equiprobable and they total to $\dfrac{1}{7}$ in probability so they are both $\dfrac{1}{14}$ in probability.
@Romsek, it clearly says THAT boy has an older brother and younger sister. So the elementary event is BBG. We cannot count GBB because it violates the given.

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#### romsek

MHF Helper
@Romsek, it clearly says THAT boy has an older brother and younger sister. So the elementary event is BBG. We cannot count GBB because it violates the given.
No. It asks what is the probability that the boy has an older brother and younger sister.

#### joshuaa

GBB could be either the "boy" having an older brother as we want, or
it could be that the "boy" is the older brother and thus has a younger brother.
Both are equiprobable and they total to $\dfrac{1}{7}$ in probability so they are both $\dfrac{1}{14}$ in probability.
I agree GBB are equiprobable, but they violate the event as Plato said

Why not saying the Event BBG could be either the "boy" having an older brother as we want, or
it could be that the "boy" is the older brother and thus has a younger brother.

then 1/14 answer makes more sense

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#### romsek

MHF Helper
I agree GBB are equiprobable, but they violate the event as Plato said

Why not saying the Event BBG could be either the "boy" having an older brother as we want, or
it could be that the "boy" is the older brother and thus has a younger brother.

then 1/14 answer makes more sense
That's what I meant.

we know that BBG has probability $\dfrac 1 7$

BBG can be either our boy has an older brother, or that he is the older brother.

Both with equal probability, and summing to $\dfrac 1 7$, i.e. $\dfrac{1}{14}$

So $P[\text{our boy has older brother and younger sister}] = \dfrac{1}{14}$

#### Plato

MHF Helper
From families with three children, a family is selected at random and found to have
a boy. What is the probability that has (a) an older brother and a younger
sister; (b) an older brother; (c) a brother and a sister? Assume that in a three-child
family all gender distributions have equal probabilities.
If I were tasked with editing this question, here is what I would submit.
In a family of with three children what is the probability that a boy who has:
A) an older brother and a younger sister?
B) an older brother?
C) a brother and a sister?

The advantage in that wording is that any hint of a conditional is removed.