Needhelp on a) Find the discrete probability distribution for x, the number lemons selected. the rest i can do

Suppose that of the next 12 cars to come off the assembly line in a large automobile company,


three (3) are known to be lemons (lots of problems). Two of the twelve are chosen at random.


a) Find the discrete probability distribution for x, the number lemons selected.


b) Construct a discrete probability histogram for x, the number of lemons selected.


c) Calculate the expected number of ‘lemons’ chosen for this sample.


d) Calculate the standard deviation for the number of ‘lemons’ chosen.

Needhelp on a) Find the discrete probability distribution for x, the number lemons selected. the rest i can do
Number of ways to get 0 lemons in a sample of 2


=Choose(9,2)=36


Number of ways to get 1 lemon in a sample of 2


=Choose(3,1)*Choose(9,1)=27


Number of ways to get 2 lemons in a sample of 2


=Choose(3,2)=3


Number of possible samples of 2


=Choose(12,2)=66


Probability of getting 0 lemons in a sample of 2


=Choose(9,2)/Choose(12,2)=6/11


Probability of getting 1 lemon in a sample of 2


=Choose(3,1)*Choose(9,1)


/Choose(12,2)=9/22


Probability of getting 2 lemons in a sample of 2


=Choose(3,2)/Choose(12,2)=1/22





See distribution at





http://i25.tinypic.com/21n0ead.jpg





where Choose(a,b)=a!/b!/(a-b)!





Expected lemons=0*6/11+1*9/22+2*1/22


=1/2=.5





Expected lemons^2=0^2*6/11+1^2*9/22


+2^2*1/22=13/22





Variance=(Expected lemons^2)-(Expected lemons)^2


=(13/22)-(1/2)^2=15/44





Standard Deviation=sqrt(Variance)


=1/22*sqrt(165)=.5838742081
Reply:It's Binomial.





X~Bin(n, p)


n = 2, p = 3/12 = 0.25





X~Bin (2, 0.25)
Reply:This is NOT a binomial distribution as I first thought, and as two of the answerers stated. A binomial distribution requires the trials be independent, but in this question the probability on the second choice depends on the result of the first choice.





ksoileau has the correct distribution and the derivation for expectation and variance. (with a small glitch on the last line. It should be sqrt(15/44) =.5838742081)





I will show another way to get the distribution, but please also learn the method that ksoileau used as it is more powerful.





P(lemons = 0) = (9/12)(8/11) = 6/11


P(lemons = 1) = (3/12)(9/11) + (9/12)(3/11) = 9/22


P(lemons = 2) = (3/12)(2/11) = 1/22





To get zero lemons you must choose a non-lemon on the first choice, p = 9/12, and a non-lemon on the second choice p = 8/11. There would be 8 non-lemons left and 11 cars left for the second choice if a non-lemon was chosen the first time.





To get one lemon you must chose EITHER a lemon the first time, p = 3/12 and a non-lemon the second time, p = 9/11; OR a non-lemon the first time, p = 9/12 and a lemon the second time, p = 3/11.





To get two lemons you must chose a lemon the first time, p = 3/12 and a lemon the second time, p = 2/11.








To get the expected number of lemons; multiply the number of lemons by the probability and add, i.e.


0(6/11) + 1(9/22) + 2(1/22).





To get the standard deviation first find the variance and then take the square root.


To get the variance; multiply the square of the number of lemons by the probability and add, and then subtract the square of the expectation. i.e.


0(6/11) + 1(9/22) + 4(1/22) - (1/2)^2 = 13/22 - 1/4 = 15/44.





Standard deviation = sqrt(15/44)





If you like this method, please, choose ksoileau's answer as best. He got it correct from the beginning while the rest of us were thinking "binomial."
Reply:Probability of an item being a lemon is 3/12=1/4


2 are chosen


We want the probability that 0 are lemon, 1 is lemon, both 2 are lemons.


x--p(x)


0 --0.5625


1 --0.375


2 --0.0625


This is how you get the probabilities.


Use a Binomial distribution with n=2 p=0.25 (1/4) and x=0,1,2


P(x=k)=nCk p^k (1-p)^(n-k)


where nCk = n! / k! (n-k)!


You'll see that (after you finish your question)


The expected number = np = 2(0.25)=1/2


variance = np(1-p) = 2(1/4)(3/4)=6/16


i.e. standard deviation is sqrt(6/16)