Chapter 6.3, Problem 24P

Expand Your Knowledge: Geometric Distribution; Agriculture

Approximately $3.6\%$ of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Source: Australian Journal of Agricultural Research. Vol. 25. pp. 783-790). (Bitter pit is a disease of apples resulting in a soggy core, which can be caused either by overwatering the apple tree or by a calcium deficiency in the soil.) Let n be a random variable that represents the first Jonathan apple chosen at random that has bitter pit.

(a) Write out a formula for the probability distribution of the random variable n.

(b) Find the probabilities that $n=3.n=5.andn=12.$

(c) Find the probability that $n\ge 5.$

(d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use $\mu$ for the geometric distribution and round.

Hint: See Problem 23.