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