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 determine the probability that all 5 members selected for the committee are girls, we'll use combinations to calculate the number of favorable outcomes and the total possible outcomes.

**Given:**
- **Number of girls:** 13
- **Number of boys:** 10
- **Total students:** 13 girls + 10 boys = 23 students
- **Committee size:** 5

**Step 1: Calculate the Number of Ways to Choose 5 Girls**

The number of ways to choose 5 girls out of 13 is given by the combination formula:

$$
\binom{13}{5} = \frac{13!}{5!(13-5)!} = \frac{13!}{5! \cdot 8!} = 1,287
$$

**Step 2: Calculate the Total Number of Ways to Choose Any 5 Students**

The total number of ways to choose 5 students out of 23 is:

$$
\binom{23}{5} = \frac{23!}{5!(23-5)!} = \frac{23!}{5! \cdot 18!} = 33,649
$$

**Step 3: Determine the Probability**

The probability $$ P $$ that all 5 committee members are girls is the ratio of the favorable outcomes to the total possible outcomes:

$$
P = \frac{\binom{13}{5}}{\binom{23}{5}} = \frac{1,287}{33,649}
$$

**Step 4: Simplify the Fraction**

To simplify $$ \frac{1,287}{33,649} $$:

1. Find the Greatest Common Divisor (GCD) of 1,287 and 33,649, which is **11**.
2. Divide both numerator and denominator by 11:

$$
\frac{1,287 \div 11}{33,649 \div 11} = \frac{117}{3,059}
$$

Thus, the simplified probability is:

$$
\boxed{\dfrac{117}{3059}}
$$
The probability that all 5 committee members are girls is $$ \frac{117}{3059} $$.

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