Doug Kerr
Well-known member
We may read that a new floodwall has been designed to withstand "a 100-year flood". Just what is a "100-year flood"?
A common description is that this is "a flood that we would expect to occur every 100 years". But what exactly does that mean? A little thought will reveal that this isn't really a very specific definition. Would it mean that if we had such a flood in 2015, there would probably be one in 2115? Or just what?
Scientists may speak of a "p=0.01" flood. That means that in any year, the probability of a flood that great or greater occurring once or more times is 0.01. And it is such a flood that is sometimes described as a "100-year flood", although that makes the scientists shudder.
Lets assume that this probability is independent between years. That means if we had a flood that great in 2015, the probability that we would have (another) one in 2016 is still 0.01. Often people think that is not intuitive. "Well, if we had one in 2015, it's pretty unlikely we would have one in 2016". True. Only 0.01 - just like for 2015.
A useful statistical measure is the expectation of an event. That refers what we would "expect" to be the average number of occurrences of the event per time period averaged over "all time periods".
Suppose that we consider our time period to be 100 years, and we imagined a 100-year period occurring thousands of times (but with no change in the probability of a flood of a certain magnitude occurring - obviously we have to consider many "parallel universes" to imagine such a thing, but that doesn't hurt the technical concept). The average number of such floods that occur in these thousands of 100 year periods is said to be the expectation of the flood occurring within any 100-year period. It is not a measure of "how likely" but rather a measure of "how many".
And in fact, given our stipulation that the probability of such a flood occurring in a given year is independent between years, then for our "p=0.01" flood, the expectation of the number of floods of at least that size occurring in any given 100-year period is - one. It is 0.01 times 100.
A quite different question is this: "For any given 100-year period, what is the probability that there will occur at least one such flood?"
And in fact that probability is 0.634 (63.4%). Wow. How would we reckon that?
Well, if the probability that there will be at least one flood of the stated size or greater in a year is 0.01, then the probability that there will be zero such floods in a given year is 0.99.
Again, if that is independent between years, then the probability that there will be zero floods in every one of 100 years (and thus zero floods in the entire 100-year period) is just:
Now the probability that:
• this is not so; that is,
• it is not so that there is no such flood in that period; that is,
• there is at least one such flood in that period
is just 1 minus that, or 0.634 (63.4%).
So, for a large number of (identical) 100-year periods, only in about 63% of them would we expect to have a flood of this kind - what is often (but unwisely) called a "100-year flood".
Best regards,
Doug
A common description is that this is "a flood that we would expect to occur every 100 years". But what exactly does that mean? A little thought will reveal that this isn't really a very specific definition. Would it mean that if we had such a flood in 2015, there would probably be one in 2115? Or just what?
Scientists may speak of a "p=0.01" flood. That means that in any year, the probability of a flood that great or greater occurring once or more times is 0.01. And it is such a flood that is sometimes described as a "100-year flood", although that makes the scientists shudder.
Lets assume that this probability is independent between years. That means if we had a flood that great in 2015, the probability that we would have (another) one in 2016 is still 0.01. Often people think that is not intuitive. "Well, if we had one in 2015, it's pretty unlikely we would have one in 2016". True. Only 0.01 - just like for 2015.
A useful statistical measure is the expectation of an event. That refers what we would "expect" to be the average number of occurrences of the event per time period averaged over "all time periods".
Suppose that we consider our time period to be 100 years, and we imagined a 100-year period occurring thousands of times (but with no change in the probability of a flood of a certain magnitude occurring - obviously we have to consider many "parallel universes" to imagine such a thing, but that doesn't hurt the technical concept). The average number of such floods that occur in these thousands of 100 year periods is said to be the expectation of the flood occurring within any 100-year period. It is not a measure of "how likely" but rather a measure of "how many".
And in fact, given our stipulation that the probability of such a flood occurring in a given year is independent between years, then for our "p=0.01" flood, the expectation of the number of floods of at least that size occurring in any given 100-year period is - one. It is 0.01 times 100.
A quite different question is this: "For any given 100-year period, what is the probability that there will occur at least one such flood?"
And in fact that probability is 0.634 (63.4%). Wow. How would we reckon that?
Well, if the probability that there will be at least one flood of the stated size or greater in a year is 0.01, then the probability that there will be zero such floods in a given year is 0.99.
Again, if that is independent between years, then the probability that there will be zero floods in every one of 100 years (and thus zero floods in the entire 100-year period) is just:
0.99^100
which is 0.366.Now the probability that:
• this is not so; that is,
• it is not so that there is no such flood in that period; that is,
• there is at least one such flood in that period
is just 1 minus that, or 0.634 (63.4%).
So, for a large number of (identical) 100-year periods, only in about 63% of them would we expect to have a flood of this kind - what is often (but unwisely) called a "100-year flood".
Best regards,
Doug