February 18, 2013

Misunderstood Probablility

Probability of precipitation is given on virtually every weather forecast, but many people don't really understand what it is and where it comes from.

Let's test your knowledge of this important weather concept.

A probability of precipitation of 30% means:

(1)  One expects precipitation for 30% of the time.
(2)  One expects precipitation over 30% of the area.
(3)   There is a 30% chance of precipitation at some point over a particular period.
(4)  That it will precipitate 30% of the time over 30% of the area over a particular period.

Time's up.  Write it down.  The answer is number three. Probabilities are always given for a point in space over a standard period (most frequently over 12 hr time chunks).  So at that location over the specified period for similar weather conditions, we would expect it to precipitate 3 out of 10 times.
If you got it wrong don't worry about it....a lot of people do.   Consider the study by some European scientists in the journal Risk Analysis (found here).  They questioned folks in Amsterdam, Athens, Berlin, Milan, and New York about what a “30% chance of rain tomorrow” meant.  Only in New York did the majority of the survey group supply the correct answer.  You have got to respect those New Yorkers; they walk and talk fast, but boy do they know their probabilities.  In the European cities, the preferred (and incorrect) interpretations were that it will rain tomorrow “30% of the time,” followed by “in 30% of the area.”   So much for European sophistication.

Does being precipitation soaked give Northwesterners  more insight into precipitation probabilities?   University of Washington psychology professor and expert in weather information interpretation, Dr. Susan Joslyn, has completed several studies encompassing hundreds of students to answer this very question.   As described in an article in the Bulletin of the American Meteorological Society (found here), she and her colleagues found that only roughly 50% of the sodden UW students got the right answer.  Disappointing!
One more interesting question.  When was the term probability first used in a weather forecast?  The amazing answer:  the very first forecast!

Following the signing by President Ulysses S. Grant of an authorization to establish a system of weather observations and warnings of approaching storms, on February 19, 1871, Cleveland Abbe issued the first “official” public Weather Synopsis and Probabilities based on observations taken at 7:35 a.m. that day: 

"Synopsis for past twenty-four hours; the barometric pressure had diminished in the southern and Gulf states this morning; it has remained nearly stationary on the Lakes. A decided diminution has appeared unannounced in Missouri accompanied with a rapid rise in the thermometer which is felt as far east as Cincinnati; the barometer in Missouri is about four-tenths of an inch lower than on Erie and on the Gulf. Fresh north and west winds are prevailing in the north; southerly winds in the south. Probabilities: it is probable that the low pressure in Missouri will make itself felt decidedly tomorrow with northerly winds and clouds on the Lakes, and brisk southerly winds on the Gulf."

For his insistence in using the term probabilities, Cleveland Abbe was given the name "Old Probs."  In a future blog, I will describe how meteorologists come up with probabilities.  Be prepared, this is the meteorological version of sausage making.

"Old Probs" Cleveland Abbe:  The first official U.S. weather forecaster!



  1. I think the Weather Channel has to take a lot of the blame for this one. If enter my (quite small - no more than a mile across) ZIP code at weather.com, the default view is quite good, with a decent prose overview and annotations like "RIght now, locations nearby are reporting rain", but the hourly forecast is a ludicrous fake precision fest. It gives me a percentage chance of rain for every 15 minute time slice over the next few hours, without anywhere explaining that it's representative of a probability across a longer time or a wider area. And then there are the humidity and wind speed forecasts, rounded to whole numbers as if they know that the relative humidity will be exactly 79% at this roughly half-square-mile tract at midnight.

  2. Of course, an easy way for whatever forecaster, to hedge their "bet". ....

  3. Wow, I've wanted this answer for so long!

    But, I may be particularly dense, or it may be my formal training, but it's still not clear to me.

    The use of some existential and universal quantifiers would help here.

    I read the correct answer as:

    There is a point x in the zip code, or whatever area we are talking about, that is representative and for which the forecast is made. Then, assuming properly calibrated forecasts, the statement "the probability of precipitation at x in the interval t is 30%" means that in expectation, 30 out of 100 times this statement is made, there will be at least some precipitation at x during t.

    The way you wrote it, Cliff, I think it still admits the interpretation that "if you sample 100 time instants uniformly at random during t, the instantaneous weather will have some precipitation at 30 of the instants, in expectation."

  4. Having grown-up on the east coast where there's much more data available to forecast with, I'd say that there's a significant difference in what an "n% chance of precipitation" means in a given time period between here and there. Sure, in New York, it does mean that there's a n% chance that precipitation will occur at that point for the given period of time. But out here in the wet and wild Northwest, it seems more fitting when interpreted as the inverse: a 100%-n% chance of it NOT precipitating for some slice of time at the point during the forecast's range.

    I learned this the hard way when relying on point forecasts from the NWS when planning an early summer camping trip on Mt. Hood a few years ago. In true Juneuary fashion, we awoke to 4 inches of fresh snow on the ground the first morning and went on to experience consecutive days of drenching rain while the forecast called for a 10% chance of precipitation the entire time (and snow levels of over a thousand feet higher than where we were :P).

  5. Hard to believe so many people misunderstand this, i don't think its something you really have to learn its kind of just the obvious and simple way to interpret it. That said its not the best conventional forecast tool because it makes no mention of amount of rain.

  6. After trying to plan our hikes in the Olympics using weather forecasts, we finally figured that 30% rain translates to 70% of the air is water-free, with only 30% water!

  7. Here's my point of confusion: the probabilities don't add up correctly.

    I'm looking at wunderground.org's forecast for Vancouver, BC, this week. The "hourly forecast" splits the day up into eight chunks of three hours each.

    Consider Friday's forecast. Probability of precipitation for the whole day (the probability that there will be precipitation at some point during the day) is given as 80%.

    However, here are the probabilities of precipitation for each of the three-hour periods during the day:
    30%, 40%, 40% 40%, 40%, 20%, 10%, 10%.

    The probability of precipitation at at least some point during the day should be 1 - (probability of no precip at all). Probability of no precip is (prob.no.precip. in first three-hour chunk) x (prob.no.precip. in second three-hour-chunk) x etc. So if Pn is the probability of precipitation in three-hour-period n, then the probability of precipitation for the whole day should be

    1 - (1 - P1)(1 - P2) ... (1 - P8).

    However, this adds up to 94% for Friday, not the 80% that is reported.

    The only conclusion I can draw is that at least one of the probabilities of precipitation does not represent the probability that there will be some precipitation at some point during the day.

    Or wunderground is broken, but I think I've seen the same problem on other sites.

  8. The problem that I've always had with "...chance of precipitation" is that it's not very useful without specifying the type or rate of precipitation. Chance of what, a deluge of a light mist? For example, in my experience, here in the Northwest, 100% chance of precipitation does not mean you need rain gear. In other parts of the country where thunderstorms and downpours do occur, a 20% chance means you do.

    Why don't forecasters simply predict the amount (i.e. in inches) of fain to fall on an area in that time? For example, "80% chance of 0.05 in". Now that would be useful. It would mean you don't need a raincoat. Compared to "30% chance of 3 in", which would mean you do. That forecast would be far more useful. If forecasts can predict probability of precipitation, can they not also predict the rate?

    Also, I have discovered that where I live in the eastern slope of the cascades, the "...chance of precipitation" is so frequently incorrect that it is completely uninformative.

  9. I'm curious how many of the UW students were natives.

    I don't really pay attention to the percentage amount when looking at the forecast, other than to assume that if there is any percentage at all, plan for rain. In the PNW it's almost like precipitation calls out for its fellow precipitates. If there is a 10% chance of rain, those few droplets addressed in the probability feel lonely and send out a distress signal (not perceivable by meteorologists) and next thing you know, 10% chance of rain becomes 100% downpour in your microzone :)

    1. I wondered the same thing about the UW study. I bet if it was broken down by native non-native PNWers you would find the majority of correct answers came from those who grew up here.

      However as forecasting seems to ba a bit of an art so it seems interpreting a forecast is also a bit of an art. The longer you live in a given area the better your understanding of a given local forecast.

  10. Adam,

    I make the conclusion you are making an unwarranted independence assumption. :)

    We should have them draw us a little box plot for the predicted total amount of precipitation for each time interval.

  11. This is why I LOVE Probcast! There's a percentage chance, AND for each 12-hour period, there's a range given for the min and max. For instance, for tomorrow night in 98008, there is:
    a 30% chance of precip
    and a 10% chance of 0.07" or more
    The temperature is also displayed similarly...
    the low is predicted at 33F
    Chance of a freeze is 40%
    10% chance low is higher than 37F
    10% chance low is lower than 28F
    I find this very useful in the grand scheme of things... even if it does rain, it's likely to be a trivial amount.

  12. singliar, fair enough, but here's a better example:

    wunderground's forecast for Saturday shows no less than a 10% chance of precipitation in any of four six-hour periods, but a probability of 0% for the whole day. I think no assignment of conditional probabilities can make that happen.

  13. So if it is raining I forecast a decreasing probability of precipitation. If it is not raining I forecast an increasing probability of precipitation. Always verify...

    BTW you bot words are too small and a real pain.

  14. For the correct answer "There is a 30% chance of precipitation at some point over a particular period." is that point a point a specific location or a point in time. For example, if there is a 30% chance or rain in Seattle tomorrow, is that at SEATAC (or some other designated location), or anywhere in Seattle city limits?


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