Question

A city has calculated the probability of having a significant rupture in a year: p = 0.02. This probability remains fixed from year to year, independent of whether there was a rupture in a preceding year. The city has asked you to determine the following (for each question be sure to specify what probability model you are using and justify the choice.)

(a) The probability that the rupture will occur in the 4th year.

(b) The expected number of years until the first rupture.

(c) The probability of exactly 4 ruptures in the next 14 years.

(d) The standard deviation, of the number of ruptures in the next 14 years.

(e) The probability that the first rupture occurs in one of the first 4 years.

Answer 1

(a) The probability that the rupture will occur in the 4th year.

Probability = 0.98*0.98*0.98*0.02 = 0.0189

(b) The expected number of years until the first rupture.

Expected number = 1/0.02 = 50

(c) The probability of exactly 4 ruptures in the next 14 years.

Probability = 14C4*(0.02)^4*(0.98)^10 = 0.00013

(d) The standard deviation, of the number of ruptures in the next 14 years.

Standard deviation = 14*0.02*0.98 = 0.5238

(e) The probability that the first rupture occurs in one of the first 4 years.

Probability = 13C1*(0.02)*(0.98)^13 = 0.2