A dust mite allergen level that exceeds 2 micrograms per gram (g/g) of dust has been associated

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.
d. Find E(x). Give a meaningful interpretation of the result.
e. Find σ.
f. Find the exact probability that x is in the interval µ ± 2σ. Compare your answer with Chebyshev's rule and the empirical rule.