Question

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

Answer 1

Hint: We will find the probability in girls in the room A and Room B. Thereafter, we will find the girls laughed in room A and room B. Then we will apply Bayes theorem: $P\left( {\dfrac{A}{B}} \right) = \dfrac{{P\left( {\dfrac{B}{A}} \right)PA}}{{P(B)}}$

Complete step-by-step answer:
Number of boys in room A $ = 4$
Number of girls in room A $ = 2$
P(girls laugh in a room ) $ = \dfrac{1}{2}$
Total number of candidate in room A $ = 2 + 4$
Total number of candidate in room A $ = 6$
By using the formula of probability $P(E) = \dfrac{{favourable\,\,outcomes}}{{total\,\,number\,\,of\,\,outcomes}}$
P(girls laughed in room A) $ = $$\dfrac{{favourable\,\,outcome}}{{total\,\,number\,\,of\,\,outcome}}$
P(girls laughed in room A) $ = \dfrac{2}{6}$
P(girls laughed in room A) $ = \dfrac{1}{3}$
Now,
Number of girls in room B $ = 3$
Number of boys in room B $ = 5$
Total candidates in room $B = 3 + 5$
Total candidates in room B $ = 8$
Now, by using the formula of probability $P(E) = \dfrac{{favourable\,\,outcome}}{{total\,\,number\,\,of\,\,outcome}}$
P(girls laughed in room B) $ = \dfrac{3}{8}$
Now, we will apply Bay’s theorem
P(the girl who laughed was from room B) $ = \dfrac{{\dfrac{1}{2} \times \dfrac{3}{8}}}{{\dfrac{1}{2} \times \dfrac{3}{8} \times \dfrac{1}{2} \times \dfrac{1}{3}}}$
P(the girls who laughed was from room B) $ = \dfrac{{\dfrac{3}{{16}}}}{{\dfrac{3}{{16}} + \dfrac{1}{6}}}$
We will take LCM of $6,16 - 48$, we have
P(the girls who laughed was from room B) \[ = \dfrac{{\dfrac{3}{{16}}}}{{\dfrac{{3 \times 3 + 1 \times 8}}{{48}}}}\]
P(the girls who laughed was from room B) $ = \dfrac{{\dfrac{3}{{16}}}}{{\dfrac{{9 + 8}}{{48}}}}$
P(the girls who laughed was from room B) $ = \dfrac{{\dfrac{3}{{16}}}}{{\dfrac{{17}}{{48}}}}$
P(the girls who laughed was from room B) $ = \dfrac{3}{{16}} \times \dfrac{{48}}{{17}}$
P(the girls who laughed was from room B) $ = \dfrac{9}{{17}}$
Hence, the required probability is $\dfrac{9}{{17}}$


Note: Students must keep in mind that first you find the probability of girls laughed in a room .After that find the probability of girls laughed in room A and room B separately.