Question

In the western United States, there are many dry-land wheat farms that depend on winter snow and spring rain to produce good crops. About 65% of the years there is enough moisture to produce a good wheat crop, depending on the region.

(a) Let r be a random variable that represents the number of good wheat crops in n = 8 years. Suppose the Zimmer farm has reason to believe that at least 4 out of 8 years will be good. However, they need at least 6 good years out of 8 years to survive financially. Compute the probability that the Zimmers will get at least 6 good years out of 8, given what they believe is true; that is, compute P(6 ≤ r | 4 ≤ r). (Round your answer to three decimal places.)


(b) Let r be a random variable that represents the number of good wheat crops in n = 10 years. Suppose the Montoya farm has reason to believe that at least 6 out of 10 years will be good. However, they need at least 8 good years out of 10 years to survive financially. Compute the probability that the Montoyas will get at least 8 good years out of 10, given what they believe is true; that is, compute P(8 ≤ r | 6 ≤ r). (Round your answer to three decimal places.)

Answer 1

a)

here this is binomial with parameter n=8 and p=0.65

probability =

P(X>=6)=

1-P(X<=5)=

1-∑x=0x-1     (nCx)px(1−p)(n-x)    =

0.4278

probability =

P(X>=4)=

1-P(X<=3)=

1-∑x=0x-1     (nCx)px(1−p)(n-x)    =

0.8939

P(6<=r|4<=r) =P(6<=r)/P(4<=r)=0.4278/0.8939 =0.479

b)

here this is binomial with parameter n=10 and p=0.65

P(8<=r|6<=r) =P(8<=r)/P(6<=r)=0.2616/0.7515 =0.348