From a group of 10 boys and 12 girls, a committee of 4 students is chosen at random.
a. What is the probability that all 4 members on the committee will be girls?
b. What is the probability that all 4 members of the committee will be boys?
c. What is the probability that there will be at least 1 girl on the committee?

Question

UpStudy AI · Accepted Answer

To determine the probability that all 4 members of the committee are girls, we can use combinatorial methods.

**Total Number of Ways to Choose 4 Students from 22:**
There are $$10$$ boys and $$12$$ girls, making a total of $$22$$ students. The total number of ways to choose any 4 students from these 22 is given by the combination formula:

$$
\text{Total ways} = \binom{22}{4}
$$

**Number of Ways to Choose 4 Girls from 12:**
To choose 4 girls out of the 12 available, the number of ways is:

$$
\text{Ways to choose girls} = \binom{12}{4}
$$

**Probability Calculation:**
The probability that all 4 members chosen are girls is the ratio of the number of favorable outcomes (choosing 4 girls) to the total number of possible outcomes (choosing any 4 students):

$$
\text{Probability} = \frac{\binom{12}{4}}{\binom{22}{4}}
$$

**Simplified Form:**
This can also be expressed as:

$$
\text{Probability} = \frac{12 \times 11 \times 10 \times 9}{22 \times 21 \times 20 \times 19}
$$

**Final Answer:**

Problem a Answer:

The probability is $\dbinom{12}{4}$ divided by $\dbinom{22}{4}$. Thus,

Probability = $\dfrac{\binom{12}{4}}{\binom{22}{4}}$.
The probability that all 4 committee members are girls is the number of ways to choose 4 girls divided by the total number of ways to choose any 4 students. So, Probability = $\dfrac{\binom{12}{4}}{\binom{22}{4}}$.

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