Question

A 6-hour rainfall of 6 cm at a station was found to have a return period of 40 years. The probability that a 6-hour rainfall of this or larger magnitude will occur at least once in 20 successive years at that station is ______ (rounded off to three decimal places).

Hint:

Use p = 1/T for the annual exceedance probability, then P(at least once in n years) = 1 - (1-p)^n.

Answer 1

Correct Answer: 0.397

Step 1: Write down what is given. The 6-hour, 6 cm rainfall has a return period \(T = 40\) years, and we need the probability that a rainfall of this magnitude or larger occurs at least once in \(n = 20\) successive years.

Step 2: Find the annual exceedance probability. The return period and the probability of exceedance in any one year are related by \(p = \dfrac{1}{T}\). So \(p = \dfrac{1}{40} = 0.025\).

Step 3: Find the probability of the event NOT occurring in one year. The probability of the rainfall not being equalled or exceeded in a given year is \(q = 1-p = 1-0.025 = 0.975\).

Step 4: Find the probability of the event not occurring in all 20 years. Treating each year as an independent trial, the probability that the rainfall is not equalled or exceeded in any of the 20 successive years is \(q^{n} = (0.975)^{20}\). Evaluating this, \((0.975)^{20} \approx 0.603\).

Step 5: Find the probability of at least one exceedance. The probability of at least one occurrence in \(n\) years is the complement of the above, \(P = 1-q^{n} = 1-(0.975)^{20} \approx 1-0.603 = 0.397\).

Step 6: State the result. The probability that this 6-hour rainfall or larger will occur at least once in 20 successive years is approximately \(0.397\), which lies well within the accepted range of \(0.350\) to \(0.450\).