A salesperson goes door-to-door in a residential area to demonstrate the use of a new household appliance to potential customers. At the end of a demonstration, the probability that the potential customer would place an order for the product is a constant 0.2107. To perform satisfactorily on the job, the salesperson needs at least four orders. Assume that each demonstration is a Bernoulli trial.
a. If the salesperson makes 15 demonstrations, what is the probability that there would be exactly 4 orders?
b. If the salesperson makes 16 demonstrations, what is the probability that there would be at most 4 orders?
c. If the salesperson makes 17 demonstrations, what is the probability that there would be at least 4 orders?
d. If the salesperson makes 18 demonstrations, what is the probability that there would be anywhere from 4 to 8 (both inclusive) orders?
e. If the salesperson wants to be at least 90% confident of getting at least 4 orders, at least how many demonstrations should she make?
f. The salesperson has time to make only 22 demonstrations, and she still wants to be at least 90% confident of getting at least 4 orders. She intends to gain this confidence by improving the quality of her demonstration and thereby improving the chances of getting an order at the end of a demonstration.
At least to what value should this probability be increased in order to gain the desired confidence? Your answer should be accurate to four decimal places.