Question

a grocery store purchases melons from two distributors j and k. Distributor J provides melons from organic farms. Distributor K provides melons from nonorganic farms. The probability that a melon selected at random from Distributor J has a diameter greater than 137 mm is 0.2119, and the probability is 0.8413 that a melon selected at random from Distributor K will have a diameter greater than 137 mm. For all the melons at the grocery store, 70 percent of the melons are provided by Distributor J and 30 percent are provided by Distributor K. Q7. For a melon selected at random from the grocery store, what is the probability that the melon will have a diameter greater than 137 mm ? a) 0.4007 b) 0.3501 c) 0.7741 d) 0.1123 e) 0.2122 Q8. Given that a melon selected at random from the grocery store has a diameter greater than 137 mm, what is the probability that the melon will be from Distributor J ? a) 0.4561 b) 0.2341 c) 0.3701 d) 0.4451 e) 0.7819

Answer 1

The probability that a randomly selected melon from the grocery store has a diameter greater than 137 mm is 0.4007. Given that a melon selected at random from the grocery store has a diameter greater than 137 mm, the probability that the melon will be from Distributor J is 0.3701.

Step-by-step explanation:

Q7. To find the probability that a randomly selected melon from the grocery store has a diameter greater than 137 mm, we use the law of total probability. Let's denote the events as follows: A = melon from Distributor J, B = melon from Distributor K, C = melon has a diameter greater than 137 mm. The probability that a melon is from Distributor J is 0.70, and the probability that a melon is from Distributor K is 0.30. The probability that a melon from Distributor J has a diameter greater than 137 mm is given as 0.2119, and the probability that a melon from Distributor K has a diameter greater than 137 mm is given as 0.8413. Therefore, the probability that a randomly selected melon from the grocery store has a diameter greater than 137 mm can be calculated as: P(C) = P(A) * P(C|A) + P(B) * P(C|B) = 0.70 * 0.2119 + 0.30 * 0.8413 = 0.4007. Therefore, the answer is option a) 0.4007.

Q8. To find the probability that a melon selected at random from the grocery store, given that it has a diameter greater than 137 mm, is from Distributor J, we can use Bayes' Theorem. Let's denote the events as follows: A = melon from Distributor J, B = melon from Distributor K, C = melon has a diameter greater than 137 mm. We already know that P(C) = 0.4007. The probability that a melon is from Distributor J is 0.70, and the probability that a melon is from Distributor K is 0.30. The probability that a melon from Distributor J has a diameter greater than 137 mm is given as 0.2119, and the probability that a melon from Distributor K has a diameter greater than 137 mm is given as 0.8413. Therefore, the probability that a melon selected at random from the grocery store, given that it has a diameter greater than 137 mm, is from Distributor J can be calculated as: P(A|C) = (P(A) * P(C|A)) / P(C) = (0.70 * 0.2119) / 0.4007 = 0.3701. Therefore, the answer is option c) 0.3701.