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?

First, let's break it down step by step! a. To find the probability that all 4 members on the committee will be girls, we need to calculate the number of ways to choose 4 girls from the 12 available. This can be done using combinations: \[ \text{Number of ways to choose 4 girls} = \binom{12}{4} = \frac{12!}{4!(12-4)!} = 495 \] Next, we calculate the total number of ways to choose any 4 students from the group of 22 students (10 boys + 12 girls): \[ \text{Total ways to choose 4 students} = \binom{22}{4} = \frac{22!}{4!(22-4)!} = 7315 \] So, the probability that all 4 committee members are girls is: \[ P(\text{all girls}) = \frac{\binom{12}{4}}{\binom{22}{4}} = \frac{495}{7315} \approx 0.0677 \] (Would be about 6.77%) --- b. To find the probability that all 4 members of the committee will be boys, we use a similar approach: \[ \text{Number of ways to choose 4 boys} = \binom{10}{4} = \frac{10!}{4!(10-4)!} = 210 \] Using the previously calculated total number of ways to choose 4 students (which is still 7315), we find: \[ P(\text{all boys}) = \frac{\binom{10}{4}}{\binom{22}{4}} = \frac{210}{7315} \approx 0.0287 \] (Would be about 2.87%) --- c. To calculate the probability that there is at least 1 girl in the committee, it's easier to use the complement rule. First, we can find the probability of having no girls (which would mean all boys): \[ P(\text{no girls}) = P(\text{all boys}) = \frac{\binom{10}{4}}{\binom{22}{4}} \approx 0.0287 \] Now we can find the probability of at least 1 girl: \[ P(\text{at least 1 girl}) = 1 - P(\text{no girls}) = 1 - 0.0287 \approx 0.9713 \] (Would be about 97.13%) --- In summary: - Probability all members are girls: ~6.77% - Probability all members are boys: ~2.87% - Probability at least 1 member is a girl: ~97.13%