Question

A group of boys and girls know either English or Hindi. The number of boys and girls are in the ratio 1:4 given 30 % of girls know Hindi and the rest of them know English. On the other hand, 50 % of the boys know Hindi and the rest of them know English. A student chosen at random from the group of student knows Hindi, the probability that the chosen student was a girl is:
(a) \[\dfrac{12}{17}\]
(b) \[\dfrac{5}{17}\]
(c) \[\dfrac{12}{50}\]
(d) \[\dfrac{1}{10}\]

Answer 1

Hint:First of all assume the number of girls and boys by using the ratio 1:4. Now, find the number of girls and boys who speak Hindi and English individually. Finally, find the required value of the probability by using \[\dfrac{\text{Number of girls who speak Hindi}}{\text{Total students who speak Hindi}}\].

Complete step-by-step answer:
We are given that a group of boys and girls know either English or Hindi. The number of boys and girls are in the ratio 1:4 given 30 % of girls know Hindi and the rest of them know English. On the other hand, 50 % of the boys know Hindi and the rest of them know English. If a student is chosen at random from the group of students who know Hindi, we have to find the probability that the chosen student was a girl.
We are given that the number of boys and girls are in the ratio 1:4. So, let us assume the number of boys and girls as x and 4x respectively. We are given that 30 % of the girls know Hindi and the rest of them know English. So, we get,
Number of girls who know Hindi = 30 % of Total girls
\[\text{=}\dfrac{30\times 4x}{100}\]
\[=1.2x.....\left( i \right)\]
Also, the number of girls who know English = Total girls – Girls who speak Hindi
= 4x – 1.2x
= 2.8x…..(ii)
We are also given that 50 % of the boys know Hindi and the rest of them know English. So, we get,
Number of boys who know Hindi = 50 % of Total boys
\[=\dfrac{50x}{100}\]
\[=\dfrac{x}{2}\]
= 0.5x…..(iii)
Number of boys who know English = Total Boys – Boys who speak Hindi
\[=x-\dfrac{x}{2}\]
\[=\dfrac{x}{2}\]
= 0.5x…..(iv)
Now, we are given that a student is chosen at random from the group of students who know Hindi. So, we get,
Total outcomes of sample space = Girls who speak Hindi + Boys who speak Hindi
By substituting the value of girls and boys who speak Hindi from equation (i) and (iii), we get,
Total outcomes or sample space = 1.2x + 0.5x = 1.7x …..(v)
Our favorable outcome is that the chosen student from the sample space is girls. So, we get,
Favorable Outcomes = Number of girls who speak Hindi
By substituting the value of the number of girls who speak Hindi from equation (i), we get,
Favorable outcomes = 1.2x
We know that,
Probability of an event \[=\dfrac{\text{Favorable Outcomes}}{\text{Total Outcomes}}\]
So, we get, the probability that the chosen student was a girl from the group of students who speak Hindi = \[\dfrac{\text{Total girls who know Hindi}}{\text{Total number of students who know Hindi}}\]
\[=\dfrac{1.2x}{1.7x}=\dfrac{12}{17}\]
Hence, option (a) is the right answer.

Note: In this question, many students make this mistake of taking sample space as the total students in the class including boys and girls which is wrong. Sample space means total students we consider while choosing the favorable cases and here we have considered only the students who know Hindi. So, our sample space would be boys and girls who know Hindi. So, this must be there in mind. In these types of questions, it is advised to first properly separate the favorable cases and sample space or the total cases and then only find the required probability.