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 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).

(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 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).

(c) List at least three other areas besides agriculture to which you think conditional binomial probabilities can be applied.

(d) Review item 6, conditional probability, in the summary of basic probability rules at the end of Section 4.2.

And show that in part (a),

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