2. The Computer Systems Department has eight faculty, six of whom are tenured. Dr. Vonder, the chair, wants to establish a committee of three department faculty members to review the curriculum. If she selects the committee at random:

a. What is the probability all members of the committee are tenured?
b. What is the probability that at least one member is not tenured? (Hint: For this question, use the complement rule.)

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

a. Without replacement: 6/8 * 5/7 * 4/6 = ?

b. You want probability of one or both being chosen. Either-or probabilities are found by adding the individual probabilities.

P(one) = 2/8 * 6/7 * 5/6 = ?

P (two) = 2/8 * 1/7 * 6/6 = ?

To calculate the probabilities in this scenario, we need to know the total number of faculty members and the number of tenured faculty members in the Computer Systems Department.

a. Probability that all members of the committee are tenured:
There are 8 faculty members in total, and 6 of them are tenured. Since Dr. Vonder wants to select a committee of 3 members, we can calculate the probability as follows:

Probability = (Number of ways to select 3 tenured faculty members) / (Total number of ways to select 3 faculty members)

The number of ways to select 3 tenured faculty members out of the 6 tenured faculty is calculated using combinations. The formula for combinations is:
nCr = n! / (r! * (n-r)!)

Therefore, we can calculate the probability as:
Probability = 6C3 / 8C3

Calculating these combinations:

6C3 = 6! / (3! * (6-3)!) = 20
8C3 = 8! / (3! * (8-3)!) = 56

So, the probability that all members of the committee are tenured is:
Probability = 20 / 56 = 5 / 14

b. Probability that at least one member is not tenured:
To calculate this probability, we will use the complement rule. The complement of the event "at least one member is not tenured" is the event "all members are tenured." So, we can subtract the probability of all members being tenured from 1.

Probability = 1 - Probability(all members are tenured)

Using the result from part a, we have:
Probability = 1 - 5/14 = 9/14