Question

A manufacturing process has 95 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 3% of the components are identi ed as defective, and the components can be assumed to be independent. If the manufacturer stocks 100 components, what is the probability that all orders can be filled without reordering the components? (Give an exact solution, an approximate solution)

Answer

Manufacturer stocks 100 components. And process requires filling 95 customer orders. So, without reordering at most 100-95=5 customer orders (their components) may be defective. Let `X` is random variable that denotes number of defective among 95 orders. Then probability of being defective is 0.03.

Variable `X` follows binomial distribution:

`P(X=x)=([95],[x])0.03^x(1-0.03)^(95-x)`, `0<=x<=95`.

So, we need to find

`P(X<=5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)=`

`=([95],[0])0.03^0 0.97^95+([95],[1])0.03^1 0.97^94 +([95],[2])0.03^2 0.97^93+([95],[3]) 0.03^3 0.97^92+`

`+([95],[4]) 0.03^4 0.97^91+([95],[5]) 0.03^5 0.97^90~~0.933415`.

Answer: 0.933415.