Understanding the Relationship Between Probability and Recurrence Interval

The question asks for the mathematical relationship that connects the probability (P) of an event occurring in any given period (often a year) and its recurrence interval (T).

Let's define these terms in this specific context:

• Probability (P): This is the likelihood or chance that an event of a specific magnitude will occur in any single year. For example, a flood that has a 1% chance of occurring in any given year has a probability of P = 0.01.
• Recurrence Interval (T): Also known as the return period, this is the average time (usually in years) between occurrences of an event of a specific magnitude or greater. If an event has a recurrence interval of 100 years, it means that, on average, such an event occurs once every 100 years.

These two concepts are inversely related. If an event is very likely to happen every year (high probability), the average time between its occurrences (recurrence interval) will be short. Conversely, if an event is very unlikely to happen in any single year (low probability), the average time between its occurrences (recurrence interval) will be long.

Mathematically, this inverse relationship is expressed as:

\( T = \frac{1}{P} \)

This equation shows that the recurrence interval (T) is the reciprocal of the probability (P) of the event occurring in any single year.

We can rearrange this equation to find the relationship given in the options:

\( T = \frac{1}{P} \)

Multiplying both sides by P, we get:

\( PT = 1 \)

This fundamental relationship holds true when considering the probability of an event occurring in a single, independent trial (like a single year). It's a cornerstone concept in fields like hydrology, engineering, and risk assessment when dealing with extreme events like floods, earthquakes, or extreme weather.

Let's look at the options provided and see how they compare to our derived relationship:

• Option 1: PT = 1
• Option 2: PT² = 1
• Option 3: P/T = 1 (which means P = T)
• Option 4: P/T² = 1 (which means P = T²)

Comparing our derived relationship \( PT = 1 \) with the given options, we find that Option 1 matches our result.

The other options (\( PT^2 = 1 \), \( P = T \), \( P = T^2 \)) do not represent the standard inverse relationship between probability (P) of occurrence in a single year and the recurrence interval (T).

Revision Table: Probability and Recurrence Interval

Additional Information on Recurrence Interval and Probability

While \( PT = 1 \) is the fundamental relationship between the annual probability (P) and the recurrence interval (T), it's important to understand a few nuances:

• Assumption: This relationship assumes that the event occurrences are independent random events over time.
• Average Time: The recurrence interval (T) is an average. A "100-year flood" doesn't mean it happens exactly every 100 years; it means there's a 1% chance of it happening in any given year, and over a very long period, the average time between such floods would be 100 years. The event could happen twice in consecutive years or not at all for several hundred years.
• Risk over Multiple Years: The probability of an event occurring at least once over a period of 'n' years is not simply \( n \times P \). It is calculated using the binomial distribution, but a common approximation for small P is \( 1 - (1-P)^n \). For a 100-year event (P=0.01), the chance of it occurring at least once in 100 years is approximately \( 1 - (1-0.01)^{100} \approx 1 - 0.366 \approx 0.634 \) or about 63.4%. This is significantly higher than the single-year probability.
• Applications: This relationship is crucial for designing infrastructure like bridges, dams, and buildings to withstand events of a certain magnitude, based on an acceptable level of risk.

In summary, the probability (P) of an event occurring in a single year and its average recurrence interval (T) are reciprocals, leading to the fundamental relationship \( PT = 1 \).