Question

If a committee of 3 to be chosen from a group of 38 people of which you are a member. What is the probability that you will be on the committee
A. ${}^{38}{C_3}$
B. ${}^{37}{C_2}$
C. $\dfrac{{{}^{37}{C_2}}}{{{}^{38}{C_3}}}$
D. $\dfrac{{666}}{{8436}}$

Answer 1

Hint: Here we have to find the probability for the given question. As we know that probability is stated as the ratio of the number of favorable outcomes to the total number of outcomes. So, first we find total ways a committee can be formed by 38 people and then the committee is formed based on the given condition using a concept of combination. On substituting in the probability formula we obtain the required answer.

Formula Used:
${\text{Probability of event to happen P}}\left( {\text{E}} \right){\text{ = }}\dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total Number of outcomes}}}}{\text{ }}$

Complete step by step solution:
 The number of ways committee of 3 to be chosen from a group of 38 people $ = {}^{38}{C_3}$
So, Total cases $ = {}^{38}{C_3}$
If I as a member we need to choose 2 to from a group of 37 people $ = {}^{37}{C_2}$
So, Favorable outcomes $ = {}^{37}{C_2}$
By the definition of probability,
${\text{Probability of event to happen P}}\left( {\text{E}} \right){\text{ = }}\dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total Number of outcomes}}}}{\text{ }}$
$\therefore $ Probability of committee formed in which I am the member $ = \dfrac{{{}^{37}{C_2}}}{{{}^{38}{C_3}}}$

Option ‘C’ is correct

Note: Before knowing the probability, students should be aware of the concept of permutation and combination. Because to find the favorable outcomes and total number of outcomes, these concepts will be used. In most cases, if the question involves a selection process then the concept of combination will be utilized.
Combination is defined as “the arrangement of ways to represent a group or number of objects by selecting them in a set and forming the subsets”.
The formula used to calculate the combination is: ${}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$.