In eastern Colorado, there are many dry land wheat farms. The success of a sprin
ID: 3259962 • Letter: I
Question
In eastern Colorado, there are many dry land wheat farms. The success of a spring wheat crop is dependent on sufficient moisture in March and April. Assume that the probability of a successful wheat crop in this region is about 56%. So the probability of success in a single year is p = 0.56, and the probability of failure is q = 0.44. The Wagner farm has taken out a loan and needs k = 4 successful crops to repay it. Let n be a random variable representing the year in which the fourth successful crop occurs (after the loan was made).
1) Write out the formula for P(n) in the context of this application. (Use C(a,b) as the notation for "a choose b".)
2) Compute P(n = 4), P(n = 5), P(n = 6), and P(n = 7)
3) What is the probability that the Wagners can repay the loan within 4 to 7 years? Hint: Compute P(4 n 7)
4) ) What is the probability that the Wagners will need to farm for 8 or more years before they can repay the loan? Hint: Compute P(n 8). (Use 4 decimal places
5) What are the expected value and standard deviation of the random variable n?
Explanation / Answer
a) p = 0.56, and the probability of failure is q = 0.44. k = 4
P(n) = C((n-1),3)* p^4 * q^(n-4) = (n-1)C3 * 0.56^4 *0.44^(n-4)
b) P(4) = 0.56^4 = 0.0983449
P(5) = 4*0.56^4*0.44= 0.1730871296
P(6) = 10* 0.56^4 * 0.44^2 = 0.19039584256
P(7) = 20* 0.56^4 * 0.44^3 = 0.16754834145
c) P(4 n 7) = 0.0983449 + 0.1730871296 + 0.19039584256 +0.167548341
=0.62937621316
d)P(n>=8) = 1- P(n<=7) = 1 -0.62937621316= 0.37062378684 ,
note that n can not be less than 4 , n>= k {here k =4}