Question

A shipment of 100 bags of sugar is rumoured to have some bags that contain illegal drugs. Local customs officials decide to screen shipments by randomly testing 4 bags. If any are found to contain drugs, the whole consignment is seized.

Compute the probability that the shipment consisting of 95 bags of sugar and 5 bags of illegal drugs will be detected in this manner.

Answer 1

In how many ways can 4 bags be chosen from 100 bags

$$\dbinom{100}{4}$$

In how many ways can 4 bags of sugar be chosen from 96 bags of sugar

$$\dbinom{95}{4}$$

What is the probability that Local customs officials will choose 4 bags of sugar

$$P(4)=\dfrac{\dbinom{95}{4}}{\dbinom{100}{4}}=\dfrac{95!4!(100-4)!}{4!(95-4)!100!}$$

$$=\dfrac{95(94)(93)(92)}{100(99)(98)(97)}$$

The probability that the shipment consisting of 95 bags of sugar and 5 bags of illegal drugs will be detected is

$$P(be\ detected)=1-\dfrac{95(94)(93)(92)}{100(99)(98)(97)}$$

$$\approx0.188125$$