Question

question 12
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a committee of 3 people is selected randomly from a group of 5 men and 4 women.
what is the probability that the committee has at least one woman?
a. none of these
b.
c.
d.
e.
1:15:53
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Explanation:

planation:

Step1: Calculate total number of ways to form committee

The total number of people is \(5 + 4=9\). Using combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), the total number of ways to select 3 people from 9 is \(C(9,3)=\frac{9!}{3!(9 - 3)!}=\frac{9\times8\times7}{3\times2\times1}=84\).

Step2: Calculate number of ways with no women (all - men)

Number of ways to select 3 men from 5 is \(C(5,3)=\frac{5!}{3!(5 - 3)!}=\frac{5\times4}{2\times1}=10\).

Step3: Calculate probability of no - women

Probability of no - women \(P(\text{no women})=\frac{C(5,3)}{C(9,3)}=\frac{10}{84}=\frac{5}{42}\).

Step4: Calculate probability of at least one woman

Using the formula \(P(\text{at least one woman}) = 1-P(\text{no women})\). So \(P(\text{at least one woman})=1-\frac{5}{42}=\frac{42 - 5}{42}=\frac{37}{42}\approx0.881\).
Let's check the options:

• Option b: \(\frac{1}{3}\approx0.333\)
• Option c: \(\frac{5}{6}\approx0.833\)
• Option d: \(\frac{4}{9}\approx0.444\)
• Option e: \(\frac{2}{3}\approx0.667\)

Since \(\frac{37}{42}\approx0.881\) is not equal to any of the given options (b, c, d, e), the correct option is a.

Answer:

A. None of these