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.

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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.

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.

That creates a subspace. Now we just count.

a) is $\dfrac{1}{7}$. Please explain that.

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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?

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.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?

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.

I think it's $\dfrac{1}{14}$ because getting GBB could be either the "boy" having an older brother as we want, or

-------------------------

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?

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.

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.

@Romsek, it clearly saysI 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.

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

I agree GBB are equiprobable, but they violate the event as Plato saidGBB 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.

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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That's what I meant.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

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

If I were tasked with editing this question, here is what I would submit.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.

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.

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