Suppose you have binomial trials for which the probability of success on each trial is p and the probability of failure is q = 1 – p. Let k be a fi­xed whole number greater than or equal to 1. Let n be the number of the trial on which the kth success occurs. This means that the fi­rst k – 1 successes occur within the fi­rst n – 1 trials, while the kth success actually occurs on the nth trial. Now, if we are going to have k successes, we must have at least k trials. So, n = k, k + 1, k + 2, . . . and n is a random variable. In the literature of mathematical statistics, the probability distribution for n is called the negative binomial distribution. The formula for the probability distribution of n is shown in the next display.

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 65%. So, the probability of success in a single year is p = 0.65, and the probability of failure is q = 0.35. 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).

(a) Write out the formula for P(n) in the context of this application.

(b) Compute P(n = 4), P(n = 5), P(n = 6), and P(n = 7).

(c) What is the probability that the Wagners can repay the loan within 4 to 7 years? Compute P(4 £ n £ 7).

(d) What is the probability that the Wagners will need to farm for 8 or more years before they can repay the loan? Compute P(n ³ 8).

(e) What are the expected value m and standard deviation s of the random variable n? Interpret these values in the context of this application.