Let X be a random variable with the probability function36.7 A vendor of scientific instruments sells ready-inflated weather balloons with diameters of 30 cm, 50 cm, and 100 cm. Half of her customers order the 30 cm size, while 40% order the 50 cm size and 10% hire the 100 cm size.

For a balloon of diameter d cm, the amount of helium gas required to fill it is (d 3

/2000) litres, and the balloon is sold for

$0.6d.

a. Let X be the balloon diameter selected by a randomly chosen customer. Write out the probability function of X as a table.

b. What is the average amount of gas required per customer?

c. If the vendor has 20 customers per day, what is the expected amount of gas she needs for a day’s trade?

d. What is the vendor’s expected payment per customer?

e. How much payment does the vendor expect to receive in total from her next five customers?

This scheme is often used when screening for rare diseases, e.g., HIV, hepatitis, and Zika virus. It’s also used for COVID in places where it’s still rare. Another form of pooled sampling used for COVID is wastewater testing. This can identify particular buildings or sewer catchments where the infection is present.