Question

From a group of $8$ boys and $3$ girls, a committee of $5$ members to be formed. Find the probability that $2$ particular girls are included in the committee is
A) $\dfrac{4}{{11}}$
B) $\dfrac{2}{{11}}$
C) $\dfrac{6}{{11}}$
D) $\dfrac{8}{{11}}$

Answer 1

The committee will be formed by two girls and three particular members. It means the three particular members will be chosen from the remaining nine members. Then, we find the number of ways to select 2 particular girls, the number of ways to select three particular members among the 9 members and the number of ways to select five particular members among 11 members.
Finally, we can find the required probability.

Complete step-by-step answer:
It is given that; the number of boys is $8$ and the number of girls $3$. A committee of $5$ members will be formed.
We have to find the probability that $2$ particular girls are included in the committee.
So, we have,
Committee: 2 girls and 3 particular members.
The number of ways to select 2 particular girls is $^2{C_2}$.
Now, the committee has to select 3 members from $(8 + 3 - 2) = 9$ members.
The number of ways to select three particular members among rest 9 members is ${ = ^9}{C_3}$.
The number of ways to select five particular members among 11 members is ${ = ^{11}}{C_5}$.
The probability that $2$ particular girls are included in the committee is $= \dfrac{{^2{C_2}{ \times ^9}{C_3}}}{{^{11}{C_5}}}$
Simplifying we get,
$\Rightarrow \dfrac{{^2{C_2}{ \times ^9}{C_3}}}{{^{11}{C_5}}} = \dfrac{{\dfrac{{2!}}{{2!\left( {2 - 2} \right)!}} \times \dfrac{{9!}}{{3!\left( {9 - 3} \right)!}}}}{{\dfrac{{11!}}{{5!\left( {11 - 5} \right)!}}}}$
By using combination formula to solve,
$\Rightarrow \dfrac{{\dfrac{{2!}}{{2!\left( {2 - 2} \right)!}} \times \dfrac{{9!}}{{3!\left( {9 - 3} \right)!}}}}{{\dfrac{{11!}}{{5!\left( {11 - 5} \right)!}}}} = \dfrac{{84}}{{462}}$
The probability that $2$ particular girls are included in the committee is $= \dfrac{{84}}{{462}} = \dfrac{2}{{11}}$
So, the probability that $2$ particular girls are included in the committee is $\dfrac{2}{{11}}$.

$\therefore$ The correct option is B) $\dfrac{2}{{11}}$.

Note: We must keep in mind that the probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.