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
Show Hint
- \(\frac{125}{143} \)
- \(\frac{18}{143} \)
- \(\frac{60}{143} \)
- \(\frac{65}{143} \)
The Correct Option is A
Solution and Explanation
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 $\boxed{\frac{125}{143}}$.
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