# Thread: Family of Three Children - Conditional Probability

1. ## Re: Family of Three Children - Conditional Probability

Originally Posted by joshuaa
Yeah the problem (b) cares about boys, especially the older brother and (B) B B event has boys. Why do we have to exclude it?
we don't exclude it

there are 6 ways the boys can be ordered. thus each way has probability 1/42

in 4 of those orders the "boy" has an older brother.

Thus BBB contributes 2/21 to the overall probability.

There are 3 other combos with 2 boys contributing 1/14 each for a total probability of 13/42

2. ## Re: Family of Three Children - Conditional Probability

Originally Posted by romsek
This is all exactly as I would have (and did) do it. You care about the boys because the problem asked for the
probability of an older brother. Not an older sister.
(c) is correct. Any combination of 2 B's and 1 G satisfies this. There are 3 of them.
Originally Posted by joshuaa
Yeah the problem (b) cares about boys, especially the older brother and (B) B B event has boys. Why do we have to exclude it?
These two quotes prompt me to ask if either of you have had any training in formal logic?
The two logical connectives, and & or serve similar functions while being totally at odds.
$P\text{ and }Q$ means that both P & Q must be true; while $P\text{ or }Q$ means that at least one of them is true.
So if a boy has an older brother and a younger sister then that is only one case $\bf{BBG}$ There can be no other.
So if the question were: In a family of three children, what is the probability that a boy has an older brother and a younger sister? Then the answer is $\dfrac{1}{8}$.

Now if the question were: In a family of three children it is known that at least one is a boy, what is the probability that boy has an older brother and a younger sister?

3. ## Re: Family of Three Children - Conditional Probability

if (B) is the older brother

why are (B) B B, (B) B G, (B) G B, and G (B) B not considered as the boy has an older brother?

4. ## Re: Family of Three Children - Conditional Probability

Originally Posted by Plato
These two quotes prompt me to ask if either of you have had any training in formal logic?
The two logical connectives, and & or serve similar functions while being totally at odds.
$P\text{ and }Q$ means that both P & Q must be true; while $P\text{ or }Q$ means that at least one of them is true.
So if a boy has an older brother and a younger sister then that is only one case $\bf{BBG}$ There can be no other.
So if the question were: In a family of three children, what is the probability that a boy has an older brother and a younger sister? Then the answer is $\dfrac{1}{8}$.

Now if the question were: In a family of three children it is known that at least one is a boy, what is the probability that boy has an older brother and a younger sister?
Plato

here i will do this
at least one boy means
probability of one boy + probability of two boys + probability of three boys
or just 1 - probability of zero boy

which is 1 - 1/8 = 7/8

so probability is (1/8) / (7/8) = 1/7

but BBG has equiprobability
if (B) is the older brother, then
probability of (B)BG = B(B)G = 1/14

then the answer will be 1/14 because only B(B)G satisfies the condition

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