Question

(1 point) The employees of a firm that manufactures insulation are being tested for indications of asbestos in their lungs. The firm is requested to send 5 employees who have positive indications of asbestos on to a medical center for further testing. If 50% of the employees have positive indications of asbestos in their lungs, find the probability that 9 employees must be tested in order to find 5 positives. If each test costs 400 Swedish kronor, find the expected value of the total cost of conducting the tests necessary to locate the 5 positives. Swedish kronor Find the variance of the total cost necessary to locate the 5 positives.

Answer 1

Negative Binomial Formula. Suppose a negative binomial experiment consists of x trials and results in r successes. If the probability of success on an individual trial is P, then the negative binomial probability is:

^x-1C_r-1 * P^r * ( 1 - P)^x-r

In our case, x = 9

r = 5

P = 0.5

Required probability = ^x-1C_r-1 * P^r * ( 1 - P)^x-r = ^{9 - 1}C_{5 - 1} * 0.5⁵ * ( 1 - 0.5)^{9 - 5} = 70 * 0.5⁹ = 0.1367

Expected value of the total cost = 400* Mean = 400 * ( r / P) = 400 * ( 5 / 0.5) = 4000 swedish kronor

Variance of the total cost = 400² * ( r * ( 1 - P) / P²) = 400² * (5 * 0.5 / 0.5² )

= 1600000 swedish kronor²