From a group of 10 boys and 13 girls, a committee of 5 students is chosen at random.
a. What is the probability that all 5 members on the committee will be girls?
b. What is the probability that all 5 members of the committee will be boys?
c. What is the probability that there will be at least 1 girl on the committee?
a. The probability that all 5 members on the committee will be girls is
(Type an integer or a simplified fraction.)

Question

UpStudy AI · Accepted Answer

To find the probability that all 5 members of the committee will be girls, we'll use combinatorial analysis.

**Total number of students:**
- Boys: 10
- Girls: 13
- **Total:** $$10 + 13 = 23$$

**Total ways to choose a committee of 5:**
$$
\binom{23}{5} = \frac{23!}{5!(23-5)!} = 33,649
$$

**Ways to choose 5 girls from 13:**
$$
\binom{13}{5} = \frac{13!}{5!(13-5)!} = 1,287
$$

**Probability that all 5 members are girls:**
$$
\text{Probability} = \frac{\binom{13}{5}}{\binom{23}{5}} = \frac{1,287}{33,649}
$$

**Simplifying the fraction:**
- The greatest common divisor (GCD) of 1,287 and 33,649 is 11.
$$
\frac{1,287 \div 11}{33,649 \div 11} = \frac{117}{3,059}
$$

So, the simplified probability is:

$$
\boxed{\dfrac{117}{3059}}
$$
The probability that all 5 members on the committee will be girls is $$\dfrac{117}{3059}$$.

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