Question

A four member committee is to be formed from a group containing 9 men and 5 women. If a committee is formed randomly, then the probability that it contains atleast one woman is

• A. \(\frac{125}{143} \)
• B. \(\frac{18}{143} \)
• C. \(\frac{60}{143} \)
• D. \(\frac{65}{143} \)

Hint:

For "at least one" probability problems, it's often simpler to calculate the probability of the complementary event ("none") and subtract it from 1. This avoids having to sum the probabilities of multiple disjoint cases (e.g., exactly one woman, exactly two women, etc.). Always simplify fractions to their lowest terms.

Answer 1

The Correct Option is A

Step 1: Calculate the total number of ways to form the committee.
Total number of men = 9
Total number of women = 5
Total number of people in the group = $9 + 5 = 14$.
A committee of 4 members is to be formed.
The total number of ways to form a 4-member committee from 14 people is given by the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\): \[ N(\text{total}) = \binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} \] \[ = \frac{14 \times 13 \times 12 \times 11}{24} = 7 \times 13 \times 11 = 1001 \] Step 2: Calculate the number of ways to form a committee with no women (i.e., all men).
The event "at least one woman" is the complement of the event "no women". If the committee contains no women, it means all 4 members must be men. The number of ways to select 4 men from 9 men is: \[ N(\text{all men}) = \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \] \[ = \frac{9 \times 8 \times 7 \times 6}{24} = 9 \times 2 \times 7 = 126 \] Step 3: Calculate the probability of forming a committee with no women.
\[ P(\text{no women}) = \frac{N(\text{all men})}{N(\text{total})} = \frac{126}{1001} \] To simplify the fraction, we can divide both numerator and denominator by their greatest common divisor. Both are divisible by 7: \[ 126 \div 7 = 18 \] \[ 1001 \div 7 = 143 \] So, \(P(\text{no women}) = \frac{18}{143}\). 
Step 4: Calculate the probability of forming a committee with at least one woman.
The probability of an event happening is 1 minus the probability of its complement: \[ P(\text{at least one woman}) = 1 - P(\text{no women}) \] \[ = 1 - \frac{18}{143} \] \[ = \frac{143 - 18}{143} = \frac{125}{143} \] The final answer is $\boxed{\frac{125}{143}}$.