(Use Bayes Rule) In a backyard vineyard in Napa Valley with 10 grape vines in a row, if the weather works well (just right), rain in the spring and dry through the summer, the yield for each vine is distributed roughly binomial with N=800, p=.8 . In a drought the yield is binomial with N=900 and P=.6, while if the year is too wet, the yield of useful grapes per vine is N=700, P=.85. Under climate change the probability of a just right year is about .15, of a too wet year is .15, and a dry year is .7. On a just right year the wine can sell for 200 dollars/bottle, on a dry year the quality drops so it will sell for 100 dollars a bottle, on a wet year it will sell for 25 dollars a bottls. (For a Z score with absolute value greater than 5, assume the probability is 0).
a. The yield for all 10 vines was more than 6350 grapes. Given this yield:
i. What is the probability that you will be able to sell for 200 dollars a bottle? (Hint: break it down, given the information, what is the probability of 6350 or more from 10 vines, given a just right year, a dry year, or a wet year.)
ii. What is the probability that you will be selling for 100 dollars a bottle?
iii. What is the probability that you can only sell for 25 dollars a bottle?
iv. What is your expected revenue per bottle?