Question

A class has $8$ boys. A pupil is picked at random. The probability of picking a boy is $\dfrac{1}{5}$. The number of girls in the class who wear glasses is 4. If a pupil is picked at random, find the probability of picking a girl who does not wear glasses.
(A) $\dfrac{1}{{10}}$
(B) $\dfrac{2}{3}$
(C) $\dfrac{9}{{10}}$
(D) $\dfrac{7}{{10}}$

Answer 1

Hint:First of all, we find the total no. of girls in the class by applying the formula of probability, i.e., \[Probability{\text{ }} = {\text{ }}\dfrac{{Favorable{\text{ }}outcomes}}{{Total{\text{ }}no.{\text{ }}of{\text{ }}outcomes}}\]. Then, we calculate the no. of girls who do not wear glasses by subtracting the no. of girls who wear the glasses from the total no. of girls.

Complete step-by-step answer:
For a random event, the probability of an event, say p(E) is defined as the ratio of the number of favorable outcomes or chances and the total number of outcomes.
Probability of an event is given by,
\[Probability{\text{ }} = {\text{ }}\dfrac{{Favorable{\text{ }}outcomes}}{{Total{\text{ }}no.{\text{ }}of{\text{ }}outcomes}}\]
In this case, favorable outcomes is the no. of boys in the class and total no. of outcomes is the total no. of pupils (Boys + Girls) in the class.
\[\therefore Probability{\text{ }}of{\text{ }}picking{\text{ }}a{\text{ }}boy{\text{ }} = \dfrac{{Total{\text{ }}no.{\text{ }}of{\text{ }}boys{\text{ }}in{\text{ }}the{\text{ }}class}}{{Total{\text{ }}no.{\text{ }}of{\text{ }}pupils{\text{ }}in{\text{ }}the{\text{ }}class}}...…(1)\]
Given, Total no. of boys in class$ = 8$
Total probability of picking a boy = $\dfrac{1}{5}$
Let the total no. of girls in the class be $x$.
Now substituting all the values we have in equation (1),we get-
$\dfrac{1}{5} = \dfrac{8}{{8 + x}}$
$8 + x = 8 \times 5$
$8 + x = 40$
$x = 40 - 8$
$x = 32$
Hence, the total no. of girls in the class is 32.
Also given that the no. of girls in the class who wear glasses is 4.
Therefore, the no. of girls who do not wear glasses can be obtained by subtracting the no. of girls who wear the glasses from the total no. of girls.
Hence, the no. of girls in the class who does not wear glasses$ = 32 - 4$$ = 28$
Now we have to find the probability of picking a girl who does not wear glasses. For that we are using following formula,\[\;\] \[Probability{\text{ }}of{\text{ }}picking{\text{ }}a{\text{ }}girl{\text{ }}who{\text{ }}does{\text{ }}not{\text{ }}wearing{\text{ }}glasses{\text{ }} = \dfrac{{No.{\text{ }}of{\text{ }}girls{\text{ }}who{\text{ }}not{\text{ }}wearing{\text{ }}glasses}}{{Total{\text{ }}no.{\text{ }}of{\text{ }}pupils}}\]$ = $$\dfrac{{28}}{{40}}$
$ = \dfrac{7}{{10}}$

So, the correct answer is “Option D”.

Note:After calculating the total no. of girls, we can use another approach to solve this question, i.e., we find the probability of picking a girl who wear glasses and then subtract it from $1$ to calculate the probability of picking a girl who wear glasses because the sum of probabilities of happening and not happening of an event is always $1$.