A warehouse employs 22 workers on first shift and 14 workers on second shift. Eight workers are chosen at random to be interviewed about the work environment. Find the Probability of choosing:

A. All first-shift workers
B. All second-shift workers
C. Six first-shift workers
D. Four second-shift workers

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events. The numbers get smaller because there is no replacement.

A. 22/36 * 21/35 * 20/34 * 19/33 * 18/32 * 17/31 * 16/30 * 15/29 = ?

B. 14/36 * 13/35 ….

C. 22/36 * 21/35 * 20/34 * 19/33 * 18/32 * 17/31 * 14/30 * 13/29 = ?

D. 22/36 * 21/35 * 20/34 * 19/33 * 14/32 * 13/31 ….

To find the probability of choosing specific numbers of first-shift or second-shift workers, we need to calculate the total number of possible combinations of workers that can be selected and then divide it by the total number of possible combinations of workers overall.

Before we begin, it's important to note that we will assume that every worker is equally likely to be chosen for the interview, and that once a worker is chosen, they are not replaced before selecting the next worker.

To calculate the total number of possible combinations, we can use the concept of combinations, which is given by the formula:

C(n, r) = n! / (r!(n-r)!)

Where n is the total number of workers available (first-shift + second-shift), and r is the number of workers to be chosen for the interview.

Let's go through each scenario:

A. All first-shift workers:
There are 22 first-shift workers, and we need to choose 8 of them. Therefore, the probability of selecting all first-shift workers can be calculated as:
P(A) = C(22, 8) / C(22 + 14, 8)

B. All second-shift workers:
There are 14 second-shift workers, and again, we need to choose 8 of them. Therefore, the probability of selecting all second-shift workers can be calculated as:
P(B) = C(14, 8) / C(22 + 14, 8)

C. Six first-shift workers:
This time we need to choose 6 first-shift workers out of 22 and 2 second-shift workers out of 14. Therefore, the probability of selecting 6 first-shift workers can be calculated as:
P(C) = C(22, 6) * C(14, 2) / C(22 + 14, 8)

D. Four second-shift workers:
Similarly, we need to choose 4 second-shift workers and 4 first-shift workers. Therefore, the probability of selecting 4 second-shift workers can be calculated as:
P(D) = C(14, 4) * C(22, 4) / C(22 + 14, 8)

Now, you can use these formulas to calculate the probabilities for each scenario by substituting the values into the respective formulas.

To find the probabilities, we need to know the total number of workers (N) and the number of workers of interest (K).

A. Probability of choosing all first-shift workers:
Total number of workers (N) = 22 + 14 = 36
Number of first-shift workers (K) = 22
Probability = (K choose K) / (N choose K)
Probability = (22 choose 8) / (36 choose 8)

B. Probability of choosing all second-shift workers:
Number of second-shift workers (K) = 14
Probability = (K choose K) / (N choose K)
Probability = (14 choose 8) / (36 choose 8)

C. Probability of choosing six first-shift workers:
Probability = (K choose K) * (N - K choose 8 - K) / (N choose 8)
Note: (N-K choose 8-K) is multiplied because we are choosing the remaining 8 - K workers from the remaining N - K workers.
Probability = (22 choose 6) * (14 choose 8-6) / (36 choose 8)

D. Probability of choosing four second-shift workers:
Probability = (K choose K) * (N - K choose 8 - K) / (N choose 8)
Probability = (14 choose 4) * (22 choose 8-4) / (36 choose 8)

You can calculate these probabilities using the binomial coefficient formula, and simplify using factorials.