What is the probability that the girl who laughed loudly was from room number 2?

There are 4 boys and 2 girls in room number 1 and 5 boys and 3 girls in room number 2. A girl from one of the two rooms laughed loudly. What is the probability that the girl who laughed was from room number 2.

I am finding this question very confusing. One thing that I have figured out is we have to use Bayes' theorem.

If I take $$E_1, E_2$$ and $$A$$ as following:

$$E_1=$$Event in which the girl is from room number 1,

$$E_2=$$Event in which the girl is from room number 2,

$$A=$$Event in which a girl from one of the two rooms laughed loudly.

Then, we have to find $$P(E_2/A)$$

$$P(E_1)=1/7$$

$$P(E_2)=3/14$$

$$P(A/E_1)=1/3$$

$$P(A/E_2)=3/8$$

$$P(E_2/A)=\frac{P(A/E_2)P(E_2)}{P(A/E_1)P(E_1)+P(A/E_2)P(E_2)}=27/43$$

If I consider $$E_1, E_2$$ and $$A$$ as the following,

$$E_1=$$Event in the person is from room number 1,

$$E_2=$$Event in the person is from room number 2,

$$A=$$Event in which a girl from one of the two rooms laughed loudly.

then I am getting 3/5.

Which one is correct?

• Are you sure the question isn't trying to trick you into using Bayes' theorem by giving you irrelevant information (namely the number of boys, unless you've got some information about how laughter - provoking they are)? – timtfj Dec 25 '18 at 19:10
• This question was given in the exercise for Bayes' theorem. Many such questions can be solved without using the theorem directly but Bayes' theorem provides an organized approach as per my knowledge. If you use Bayes' theorem then you have to take the number of boys into account to get 3/5 which @Daniel Mathias also got. Also, you do not need information about "how laughter-provoking they are". The language of the question is a bit tricky. They are actually asking you to find the probability that a person is from room number 2 given that she is a girl. – MrAP Dec 25 '18 at 20:52

There are $$5$$ girls.
$$3$$ of the $$5$$ are in room $$2$$.
$$P=\frac{3}{5}$$