Approximately 12.6% of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin. (Bitter pit is a disease of apples resulting in a soggy ore, 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. (Use p and n in your answer.) P (n) = (b) Find the probabilities that n = 3, n = 5, and n = 12. (Use 3 decimal places.) P (3) P (5) P (12) (c) Find the probability that n ≥ 5. (Use 3 decimal places.) (d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use μ for the geometric distribution and round.

Question

Kieran Keller

Mathematics

3 October, 14:45

Approximately 12.6% of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin. (Bitter pit is a disease of apples resulting in a soggy ore, 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. (Use p and n in your answer.) P (n) = (b) Find the probabilities that n = 3, n = 5, and n = 12. (Use 3 decimal places.) P (3) P (5) P (12) (c) Find the probability that n ≥ 5. (Use 3 decimal places.) (d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use μ for the geometric distribution and round.

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Answers (1)

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Johnny Doyle · Answer 1

Step-by-step explanation:

Let n be a random variable that represents the first Jonathan apple chosen at random that has bitter pit.

a) P (X = n) = q (n-1) p, where q = 1 - p.

From the information given, probability if success, p = 12.6/100 = 0.126

b) for n = 3, the probability value from the geometric probability distribution calculator is

P (n = 3) = 0.096

For n = 5, the probability value from the geometric probability distribution calculator is

P (n = 5) = 0.074

For n = 12, the probability value from the geometric probability distribution calculator is

P (n = 12) = 0.8

c) For n ≥ 5, the probability value from the geometric probability distribution calculator is

P (n ≥ 5) = 0.58

d) the expected number of apples that must be examined to find the first one with bitter pit is the mean.

Mean = 1/p

Mean = 1/0.126 = 7.9

Approximately 8 apples