Question

You operate a non-profit foodbank that accepts food donations and packages them into meals for local families who are food insecure. You accept canned goods from groceries. Some cans are not acceptable due to a compromised can or an expired use-by label. Can donations are assembled into boxes of 50 cans each for inspection to determine which cans should be discarded. The initial screening decision sends the box to either an experienced inspector or an inexperienced inspector.

The screener looks at 4 cans in each box. If there are zero unacceptable cans, the box is sent to an inexperienced inspector. Otherwise, it is sent to an experienced inspector.

a. Assuming a rate of 8% unacceptable, what is the probability of sending a box to an experienced inspector?

b. An inexperienced inspector makes $16 an hour, and an experienced one makes $22 an hour. If you were able to convince the groceries to reduce their unacceptable rate to 4%, what percent savings would you realize?
Assume that the mix of inspector types in FTEs equals the probability of a box being sent to each type.
For example, if 50.1% of boxes go to experienced inspectors, the FTE mix is 50.1 experienced FTES and 49.9 inexperienced FTEs. You do not change the number of inspectors, just the mix.

Answer 1

Explanation:

Binomial distribution is to be used here due to following reasons.

(a)

Probability of sending a box to an experienced inspector

= Probability of getting non-zero unacceptable cans

= 1 - Probability of getting zero unacceptable cans

=1- P(X = 0) = 1 - 10.08^0 *(1 – 0.08)^(4-0) = 0.283607

(b)

Expected cost per inspector in an hour in case of 8% unacceptable cans

= {(1-0.283607)*16+0.283607*22} = $ 17.70164

If groceries reduce their unacceptable rate to 4% then X - Bin(4, 0.04) .

In this scenario,

Probability of sending a box to an experienced inspector

= Probability of getting non-zero unacceptable cans

= 1 - Probability of getting zero unacceptable cans

=1- P(X = 0) = 1 - 0.04^0 * (1 – 0.04)*(4-0) = 0.150653

Expected cost per inspector in an hour in case of 4% unacceptable cans

= {(1-0.150653)*16+0.150653*22} = $ 16.90392

Percentage of savings realized = (17.70164-16.90392)/17.70164*100% = 4.506475%