Question

Please Solve Subdivision d),e) and f) only

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 least4 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.

Answer 1

(d) n = 18, p = 0.2107

P(X = x) = $\binom{n}{x}p^{x}(1-p)^{n-x}$ , x = 0,1,2,......,18

The probability that there would be anywhere from 4 to 8 orders = P(4 ≤ X ≤ 8)

= 0.5382

(e) Let the required number of demonstrations be n

p = 0.2107

The given condition -> P(X ≥ 4) ≥ 0.90

To find the smallest n satisfying this condition

1 - P(X < 4) ≥ 0.90

-> P(X < 4) < 0.10

-> $0.7893^{n}+\binom{n}{1}*0.2107*0.7893^{n-1} +\binom{n}{2}*0.2107^{2}*0.7893^{n-2}+ \binom{n}{3}*0.2107^{3}*0.7893^{n-3} < 0.1$ -> The smallest n = 30

She should make atleast 30 demonstrations

(f) n = 22, p = ?

The given condition -> P(X ≥ 4) ≥ 0.90

The smallest p is 0.2790

This probability should be increased atleast to 0.2790