This is another question like this one. And by the same reason, the book only has the final answer, I'd like to check if my reasoning is right.
A couple has 2 children. What is the probability that both are girls if the eldest is a girl?
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$\begingroup$ I think perhaps some couples are biologically more likely to have girls than boys. The fact that the older daughter is a girl provides some (however small) evidence that this is one of those couples. Which raises the probability of the younger child being a girl to something slighty more than $.5$. $\endgroup$– littleOOct 10, 2013 at 3:35
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$\begingroup$ @littleO I see you have a point, but in these kind of exercises, I believe you can assume 0.5. $\endgroup$– srodriguexOct 10, 2013 at 13:55
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2 Answers
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The sample space is $S = \{(g,g),(g,b),(b,g),(b,b)\}$, where $b$ is for boy, $g$ is for girl the first element of the tuple is the eldest.
Let $B$ the event the eldest is a girl, so $B=\{(g,b),(g,g)\}$.
$A$ is the event where the two children are girls. $A = \{(g,g)\}$.
Then:
$$ P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{\dfrac{|A\cap B|}{|S|}}{\dfrac{|B|}{|S|}}=\frac{1}{2}$$.
The end.
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2$\begingroup$ Yes, this is correct. Alternatively, you could argue that once you know that the elder child is a girl, the relevant sample space is $S'=\{\langle g,g\rangle,\langle g,b\rangle\}$, and $$\Bbb P(A)=\frac{|A|}{|S'|}=\frac12\;.$$ $\endgroup$ Oct 10, 2013 at 3:23
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$\begingroup$ I removed my comment, because I misread the solution. $\endgroup$ Oct 10, 2013 at 3:26
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$\begingroup$ I marked this answer as the solution because it received the most up votes. $\endgroup$ Oct 19, 2013 at 13:56
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An alternative viewpoint:
For the eldest child to be a girl, they must have had a girl first. Therefore the probability of there being two girls is the probability of having a second girl which is $\frac{1}{2}$.
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$\begingroup$ I agree that the first answer might be more formally correct, but in my view this encapsulates the given information more naturally. The probability of having another girl after the first is 1/2. $\endgroup$ Oct 10, 2013 at 4:23