Suppose that a healthcare provider selects 20 patients randomly (without replacement) from among 500 to evaluate adherence to a medication schedule. Suppose that 10% of the 500 patients fail to adhere with the schedule. Determine the following:

(a) Probability that exactly 10% of the patients in the sample fail to adhere.

(b) Probability that fewer than 10% of the patients in the sample fail to adhere.

(c) Probability that more than 10% of the patients in the sample fail to adhere.

(d) Mean and variance of the number of patients in the sample who fail to adhere.