Rainfall Probability Calculation Using Return Period

This solution outlines the steps to calculate the probability of a specific rainfall event occurring at least once within a given period, using the concept of return periods.

Calculate Annual Probability (p)

The probability ($p$) of an event with return period ($T$) occurring in any single year is calculated using the formula:

$p = \frac{1}{T}$

Given the return period $T = 40$ years for the 6 cm rainfall:

$p = \frac{1}{40} = 0.025$

Calculate Probability of Non-Occurrence (q)

The probability ($q$) that the specified rainfall event does *not* occur in a single year is:

$q = 1 - p$

$q = 1 - 0.025 = 0.975$

Calculate Probability Over Successive Years

The probability that the event does *not* occur in $n$ consecutive years assumes independence between years and is calculated as $q^n$.

We need the probability for $n = 20$ successive years.

Probability of non-occurrence in 20 years = $q^{20} = (0.975)^{20}$

Calculating the value:

$(0.975)^{20} \approx 0.6026537$

Determine Probability of At Least One Occurrence

The probability that the event occurs *at least once* in $n$ years is the complement of it never occurring:

Probability (at least once in $n$ years) = $1 - q^n$

Using $n=20$:

Probability (at least once in 20 years) = $1 - (0.975)^{20}$

Probability = $1 - 0.6026537 \approx 0.3973463$

Final Answer Rounding

The question requires the answer rounded to three decimal places.

Rounding $0.3973463$ gives $0.397$.

This calculated probability of $0.397$ lies within the given range (0.35 to 0.45).