A company finds that one out of five employees will be late to work on a given day. If this company has 42 employees, find the probabilities that the following number of people will get to work on time. (Round your answers to 4 decimal places.)

(a) Exactly 33 workers or exactly 37 workers.

(b) At least 29 workers but fewer than 37 workers.

(c) More than 27 workers but at most 39 workers.

To solve these probability problems, we can use the binomial probability formula. The binomial probability formula is:

P(x) = C(n, x) * p^x * (1 - p)^(n - x),

where:
- P(x) is the probability of x successes,
- C(n, x) is the number of combinations of n items taken x at a time,
- p is the probability of success for each trial,
- (1 - p) is the probability of failure for each trial,
- n is the total number of trials.

Now let's solve each part of the question:

(a) Exactly 33 workers or exactly 37 workers.

To find the probability of exactly 33 workers getting to work on time:
- n = 42 (total number of employees),
- x = 33 (number of successes),
- p = 4/5 (probability of success for each employee getting to work on time).

Using the binomial probability formula:
P(33) = C(42, 33) * (4/5)^33 * (1 - 4/5)^(42 - 33).

To find the probability of exactly 37 workers getting to work on time:
- n = 42 (total number of employees),
- x = 37 (number of successes),
- p = 4/5 (probability of success for each employee getting to work on time).

Using the binomial probability formula:
P(37) = C(42, 37) * (4/5)^37 * (1 - 4/5)^(42 - 37).

(b) At least 29 workers but fewer than 37 workers.

To find the probability of at least 29 workers but fewer than 37 workers getting to work on time, we need to find the sum of probabilities for x = 29, 30, 31, 32, 33, 34, 35, and 36.

P(at least 29 but fewer than 37) = P(29) + P(30) + P(31) + P(32) + P(33) + P(34) + P(35) + P(36).

You can use the binomial probability formula to calculate each individual probability, similar to how we did in part (a).

(c) More than 27 workers but at most 39 workers.

To find the probability of more than 27 workers but at most 39 workers getting to work on time, we need to find the sum of probabilities for x = 28, 29, 30, ..., 39.

P(more than 27 but at most 39) = P(28) + P(29) + P(30) + ... + P(39).

Again, you can use the binomial probability formula to calculate each individual probability.

Remember to round your answers to 4 decimal places.