Frequently Asked Questions (FAQ)
About Annual Probability of Exceedance
Your map values show ground motions that have a probability of being exceeded in 50 years of 10, 5 and 2 percent.
What is the probability of their being exceeded in one year (the annual probability of exceedance)?
What is the return period of the ground motions?
Let r = 0.10, 0.05, or 0.02, respectively. The approximate annual probability of exceedance is the ratio, r*/50, where r* = r(1+0.5r). (To get the annual probability in percent, multiply by 100.) The inverse of the annual probability of exceedance is known as the "return period," which is the average number of years it takes to get an exceedance.
Example: What is the annual probability of exceedance of the ground motion that has a 10 percent probability of exceedance in 50 years?
Answer: Let r = 0.10. The approximate annual probability of exceedance is about 0.10(1.05)/50 = 0.0021. The calculated return period is 476 years, with the true answer less than half a percent smaller.
The same approximation can be used for r = 0.20, with the true answer about one percent smaller. When r is 0.50, the true answer is about 10 percent smaller.
Surprisingly, this approximation can be used to get quick approximate answers to problems like the previous FAQ, WHEN THE KNOWN PROBABILITY IS 10 PERCENT OR LESS, but the unknown probability can be very much larger.
Example: Suppose a particular ground motion has a 10 percent probability of being exceeded in 50 years. What is the probability it will be exceeded in 500 years? Is it (500/50)10 = 100 percent?
Answer: No. We are going to solve this by equating two approximations:
r1*/T1 = r2*/T2. Solving for r2*, and letting T1=50 and T2=500,
r2* = r1*(500/50) = .0021(500) = 1.05. Take half this value = 0.525. r2 = 1.05/(1.525) = 0.69. Stop now. Don't try to refine this result.
The true answer is about ten percent smaller, 0.63.
For r2* less than 1.0 the approximation gets much better quickly.
For r2* = 0.50, the error is less than 1 percent.
For r2* = 0.70, the error is about 4 percent.
For r2* = 1.00, the error is about 10 percent.
Caution is urged for values of r2* larger than 1.0, but it is interesting to note that for r2* = 2.44, the estimate is only about 17 percent too large. This suggests that, keeping the error in mind, useful numbers can be calculated.
Here is an unusual, but useful example. Evidently, r2* is the number of times the reference ground motion is expected to be exceeded in T2 years. Suppose someone tells you that a particular event has a 95 percent probability of occurring in time T. For r2 = 0.95, one would expect the calculated r2 to be about 20% too high. Therefore, let calculated r2 = 1.15.
The previous calculations suggest the equation,
r2calc = r2*/(1 + 0.5r2*)
Find r2*.
r2* = 1.15/(1 - 0.5x1.15) = 1.15/0.425 = 2.7
This implies that for the probability statement to be true, the event ought to happen on the average 2.5 to 3.0 times over a time duration = T. If history does not support this conclusion, the probability statement may not be credible.
