Question

A dust mite allergen level that exceeds 2 micrograms per gram (mg/g) of dust has been associated with the development of allergies. Consider a random sample of four homes and let Y be the number of homes with a dust mite level that exceeds 2 mg/g. The probability distribution for Y=y, based on a study by the National Institute of Environmental Health Sciences, is shown in the following table.

y 0 1 2 3 4
p(y) .09 .30 .37 .20 .04

(a) Find the probability that three or four of the homes in the sample have a dust mite level that exceeds 2 µg/g.
(b) Find the probability that fewer than three homes in the sample have a dust mite level that exceeds 2 µg/g.

Answer 1

(a) 0.24

(b) 0.76

Explanation:

We are given that let Y be the number of homes with a dust mite level that exceeds 2 mg/g. The probability distribution for Y=y is shown below ;

y P(y)

0 0.09

1 0.30

2 0.37

3 0.20

4 0.04

(a) Probability that three or four of the homes in the sample have a dust mite level that exceeds 2 µg/g = P(y = 3) + P(y = 4)

= 0.20 + 0.04 = 0.24 .

(b) Probability that fewer than three homes in the sample have a dust mite level that exceeds 2 µg/g = P(y = 0) + P(y = 1) + P(y = 2)

= 0.09 + 0.30 + 0.37 = 0.76 .