Question

Three pear farmers (Smith, Walker, and Simpson) are planning to plant new orchards in a region which is susceptible to killing frosts. An Agriculture Canada expert has advised them that they would be foolish to plant pears if the area receives a killing frost in 1 or more years out of 5. Each farmer has been keeping yearly records of early frosts in their respective potential orchards. Using Microsoft Excel answer the questions below (Show all formulas used)

a) At Smith’s place, a killing frost has occurred twice in the last 10 years. He wants to be 75% sure that future killing frosts will occur no more often than 1 year in 5.

b) At Walker’s, a killing frost has occurred twice in the last 13 years. She wants to be 80% sure that future killing frosts occur no more often than 1 year in 5.

c) At Simpson’s, a killing frost has occurred once in the last 7 years. He wants to be 90% sure that future killing frosts will occur no more often than 1 year in 5.

For each farmer, calculate the binomial probabilities associated with a favourable outcome of 0 and 1 killing frost during a 5 year period. (15 marks total)

d) Which farmer (if any) should proceed with the planting of pear trees, given their stated willingness to take risks?

Answer 1

The syntax for using the binomial distribution in Excel is given

BINOM.DIST(number_s,trials,probability_s,cumulative)

The BINOM.DIST function syntax has the following arguments:

• Number_s: Required. The number of successes in trials.

• Trials: Required. The number of independent trials.

• Probability_s: Required. The probability of success on each trial.

• Cumulative: Required. A logical value that determines the form of the function. If cumulative is TRUE, then BINOM.DIST returns the cumulative distribution function, which is the probability that there are at most number_s successes; if FALSE, it returns the probability mass function, which is the probability that there are number_s successes.

a) Here the probability of occurrence of frost in a year is 0.2, thus the probability of no more than 1 in a year is:

BINOM.DIST(1, 5, 0.2, TRUE)= 0.73728

b) Here the probability of occurrence of frost in a year is 2/13, thus the probability of no more than 1 in a year is:

BINOM.DIST(1, 5, 2/13, TRUE)= 0.8281

c) Here the probability of occurrence of frost in a year is 1/7, thus the probability of no more than 1 in a year is:

BINOM.DIST(1, 5, 1/7, TRUE)= 0.8482

d) Walker' proceed with the planning of pear trees, as it satisfies the willingnes to take risk.