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Question:You have asked a woman if she has any children. She answered that she has two children. Then you have asked her whether if she has daughter out of those two. She said yes. What is the probability that both of her child are girls?

One answered 1/3 because one thinks that there are 3 outcomes:

boy and girl

girl and boy

girl and girl

What is the correct answer? And if you think the answer isn't 1/3, how do you explain that the above explanation is wrong?

MathsMy
  • 911
  • The answer and the explanation are correct. But it should probably be mentioned that the three situations are equally likely. – Kitegi Oct 02 '15 at 14:34
  • @Farnight isn't the answer 1/2 because 2 children are already born – MathsMy Oct 02 '15 at 14:35
  • No. If the question were "What is the probability that both of her children are girls, given that child n°1 is a girl?" or "What is the probability that both of her children are girls, given that child n°2 is a girl?", then the answer would be 1/2. You can get the same answer (1/3) using Bayes theorem, or maybe even writing a script and test it with a large number of random children. The reason most people get confused (including me. I saw this problem just a few months ago) is just because of the way we interpret the language used in the problem. – Kitegi Oct 02 '15 at 14:48
  • Related: https://math.stackexchange.com/questions/15055/in-a-family-with-two-children-what-are-the-chances-if-one-of-the-children-is-a – Henry Jun 01 '20 at 09:38

2 Answers2

1

you are quite right, $S=\{gg, gb,bg\}$(since she has said she has atleast one girl then bb doesnt happen,both are girls =$A=\{gg\}$ thus answer is $p(A)=\frac{n(A)}{n(S)}=\frac{1}{3}$.

R.N
  • 4,318
-1

Your answer is correct. You can also work it out using the formula for conditional probability:

\begin{eqnarray} P(\textrm{two girls}\ |\ \textrm{at least one is a girl}) & = & \frac{P(\textrm{two girls } \textbf{and} \textrm{ at least one is a girl})}{ P(\textrm{at least one girl})} \\ & = & \frac{\frac{\left|\{GG\}\right|}{\left|\{BB, BG, GB, GG\}\right|}}{\frac{\left|\{GB, BG, GG\}\right|}{\left|\{BB, BG, GB, GG\}\right|}}\\ & = & \frac{\frac{1}{4}}{\frac{3}{4}} \\ & = & \frac{1}{3} \end{eqnarray}

Raj
  • 964