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June 2nd, 2018, 02:32 PM   #1
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interesting probability question

Newly borned babies with probability of 0.51 are boys, girls with probability of 0.49 and gender is independent across births.

You meet a couple who tell you they have two children, and you ask whether they have any daughers. If they do (i.e. they have one or two girls), what is the probability that both of their children are girls?

Hints: The probability of two girls is 0.49^2, but what I am looking for here is the probability of two girls conditional on having at least one girl (i.e. not having two boys). Since the probability of having two boys is 0.51^2, the probability of at least one girl is 1 - 0.51^2. Can you combine these pieces of information to get the correct answer?

I am confused.. What exactly is the question asking for?? I thought the answer should have been 1 - 0.51^2

Last edited by zollen; June 2nd, 2018 at 03:32 PM.
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June 2nd, 2018, 03:58 PM   #2
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$P[\text{both children are girls | they have two children and at least one daughter}]P[\text{they have two children and at least one daughter}] = P[\text{they have two children and at least one daughter | they have two daughers}]P[\text{they have two daughers}]$

$P[\text{they have two childeren and at least one daughter|they have two daughers}] = 1$

$P[\text{they have two daughters}] = (0.49)^2 = 0.2401$

$P[\text{they have two children and at least one daughter}] = \dbinom{2}{1}(0.49)(0.51) + \dbinom{2}{2}(0.49)^2 = 0.7399$

$P[\text{both children are girls | they have two children and at least one daughter}] = \dfrac{0.2401}{0.7399} \approx 0.3245$
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June 3rd, 2018, 10:01 AM   #3
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Did you mean the Probability of [ both children are girls or they have two children and at least one daughter ]??

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$P[\text{both children are girls | they have two children and at least one daughter}] = \dfrac{0.2401}{0.7399} \approx 0.3245$
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June 3rd, 2018, 10:42 AM   #4
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Quote:
Originally Posted by zollen View Post
Did you mean the Probability of [ both children are girls or they have two children and at least one daughter ]??
no, the vertical line means given.

$P[A | B]$ means the probability of $A$ given that $B$ has occurred.
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June 3rd, 2018, 11:02 AM   #5
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It mades more sense now.
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June 3rd, 2018, 11:06 AM   #6
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It mades more sense now.
It's a direct application of Baye's Rule.
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