Question

Kamala, the student affairs director at West Moore University, has a list of student volunteers and their majors. The students on the list consist of: 10 psychology majors, 11 engineering majors, and 9 business majors. Three students are randomly selected from the list.

Find each probability. Give results accurate to 4 decimal places. Find the probability all three of the students have the same major.
Find the probability that at least one of the three students is an engineering major.
Find the probability that either all of the three students are psychology majors or none of the students are psychology majors.

Answer 1

a) The probability that all three of the randomly selected students have the same major is
`\( (10)/(30) * (9)/(29) * (8)/(28) \approx 0.0915 \).`

b) The probability that at least one of the three students is an engineering major is
`\( 1 - (19)/(30) * (18)/(29) * (17)/(28) \approx 0.3770 \).`

c) The probability that either all three students are psychology majors or none of the students are psychology majors is
`\( (10)/(30) * (9)/(29) * (8)/(28) + (20)/(30) * (19)/(29) * (18)/(28) \approx 0.3299 \)`.

Step-by-step explanation:

a) To find the probability that all three students have the same major, we calculate the probability for each major separately and then multiply them together. For psychology majors, it's
`( (10)/(30) * (9)/(29) * (8)/(28) \))`. Similarly, we do the same for engineering and business majors. Adding these probabilities gives us the overall probability, which is approximately 0.0915.

b) The probability that at least one student is an engineering major is equal to 1 minus the probability that none of them are engineering majors. So, it's
`\( 1 - (19)/(30) * (18)/(29) * (17)/(28) \)`, resulting in approximately 0.3770.

c) The probability of either all three students being psychology majors or none of them being psychology majors is the sum of the probabilities for these two mutually exclusive events. So, it's
`\( (10)/(30) * (9)/(29) * (8)/(28) + (20)/(30) * (19)/(29) * (18)/(28) \)`, which is approximately 0.3299. This accounts for the probability of having all psychology majors and none being psychology majors.

Answer 2

The probability all three of the students have the same major is 0.0909.

The probability that at least one of the three students is an engineering major is 0.7613.

The probability that either all of the three students are psychology majors or none of the students are psychology majors is 0.0909.

Based on the calculations, here are the probabilities for the given scenarios, accurate to four decimal places:

1. Probability that all three of the students have the same major:

- This probability is approximately 0.0909.

2. Probability that at least one of the three students is an engineering major:

- This probability is approximately 0.7613.

3. Probability that either all of the three students are psychology majors or none of the students are psychology majors:

- This probability is also approximately 0.0909.

These probabilities are calculated based on the combination formula, considering the total number of students in each major and the total number of students overall.