Dust mite allergies. A dust mite allergen level that exceeds 2 micrograms per gram (μg/g) of dust has been associated with the development of allergies. Consider a random sample of four homes, and let x be the number of homes with a dust mite level that exceeds 2μg/g. The probability distribution for x, based on a study by the National Institute of Environmental Health Sciences, is shown in the following table:
a. Verify that the probabilities for x in the table sum to 1.
b. Find the probability that three or four of the homes in the sample have a dust mite level that exceeds 2μg/g.
c. Find the probability that fewer than two homes in the sample have a dust mite level that exceeds 2μg/g.