Question

In an back yard vineyard in Napa valley with 10 grape vines (just right), rain in the spring and dry through summer, roughly binomial with N=800, P=.6. In a drought the yield is Bin while if the year is too wet, the yield of useful grapes per vine change the probability of a just right year is about. on a just right year the wine can sell for 100 dollars/bottle, on a dry ye will sell for 60 dollars a bottle, on wet year it will sell for 30 dollars absolute value >5 assume the probability is 0) a. The yield for all 10 vines was more than 4810 grapes. Given i. What is the probability that you will be pe vines in a row, if the weather works well gh summer, the yield for each vine is distributed ht the yield is Binomial with N=700 and P=.65, grapes per vine is N=600, P=.7. Under climate ant year is about.2, of a too wet vear is.1, and a dry year is.l. rs/bottle, on a dry year the quality drops so it it will sell for 30 dollars a bottle. (For a Z score with was more than 4810 grapes. Given this yield: is the probability that you will be able to sell for 100 dollars a bottle? (Hint, Break it down, given the information, what is the pre a just right year, a dry year, or a wet year.) the information, what is the probability of 4810 or more from 10 vines, given ii. What is the probability that you will be selling for 60 dollars a bottle? ill. What is the probability that you can only sell for 30 dollars a bottle? IV. What is your expected revenue per bottle
Please use BAYES rule! The other examples aren't showing this problem sovled using Bayes rule.

solve using Bayes rule only! no z-scores allowed.

Answer 1

Solution:

Given that, The yield of for each vine is distributed goughly binomial with N = 900 P = 0.7 In a drought the yield is binomial with N=800 P=0.75 @ A battle as sold foor 1Go when weather is right. 1. For all 10 vine gield = 6310 foar Each wine , yield =6310 - 631 let X be the yield of vine yard. - We have to find, P (x 3631 and olight weather) a PCX 3631 | Bright weather)* - P(alight weather) PC oright weather) = 0.3 P ( x 3631 I right year)

. We know that for a right year X Sem Binomial (N=900, P=0.7) using Normal approximation to Binomial 1 x follows Normal with mean Np: 900 *0.7 NP - 630 .. A Standard Deviation = ne(1-) = 13.75 NOW z = (x-630)/13.75 | sim M(0,1) P (x > 631 ( oright year) = P CZ > (631 - 630/13-75)) = p (270.07) -0.4721 ( Z-table ) P (x>631.and right. weather) = P(x8631 lought weathed)* P( right weather) = 0.4721. *0:3 -0.1416 The probability that wine is sold for $ 160 - is 0.1416

$ 100 when ( a bottle is sold foot weatheor is dory We have to find, P(X7631 and dory weather) NOW, P(X7631 and dory weatheon) - P(x631 | dary weathew) * P (dory weathear) P (dory weathens ) =0.5 Now, we have find p (xx631 I dary year) we know that for a day year xl sim Binomicel (N=800, P=0.75) using normal approximal 10 binomial & follow normal with mean NP = 800 * 0.75 = Goo and Standaod devicelion = 'Np (1-) - 12.25 then z =(x-600) / 12.25 l SimN (011) P(x>631 dory weather) = p (z >(631-600/12:25)) - P(Z > 2.53) 0.0057

P (x>631 and day weathe I and day weatheon) = P(x>631 | dory weathear ) * P(dory weather) = 0.0057 *0.5 = 0.0029 ,: The poro bo bility that wine is sold for $100 is 0.0029. 1 A bottle is sold for $30 when weather is went we have to find P(x 3631 and wet weather ) - PCX7631 | wet weather ) * pcwee weathen) p(wet weather ) -0,2 P (x) 6311 wet year) x 1 Sim Binomial (N = 700, P=0.7) using Normcel apporo rů meition to Binamiai X follow ivormal with mean Np - 700 *0.7 N P = 490 standaoid Devialion - NP (-) = 12.12 į - (x-490)/12.12 | sim Noil) p(x7631 wet weathear) = P(Z > (631-490)12.12)) = P(Z > 11063)

then P ( x x6 31 and wer weather) - P (x>631 wet weather )*e (wer weather) = 0.0000 *0.2 = 0.0000 . The probability that wine is soles for $ 300 is 0.0000. so A bottle is sold foor - $160 with probability 0.1416 $ 100 with porobability 0.0029 $ 30 with porobability 0.0000. Expected revenual pear bottle at - 160 * 0.1416 + 100-X 0.0029 + 30 *0 - $ 22.95. i Expected revenuat pear bottle is $22.95