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Looking at the National Weather Service forecast for daytime precipitation tomorrow, it shows a 70% chance of rain ("showers likely"). I interpret this as meaning that on average it will rain 7 out of 10 days with a similar forecast -- leaving 3 days of 10 with no rain.

Looking at the hourly breakdown of the same forecast it shows a 50% to 70% of rain for each daytime hour (a 12 hour period). Simplifying to 50%... if I flipped a coin at the beginning of each hour, the odds of a heads are the same as the odds for rain (50%).

Taking this further, that means that "no rain today" is as likely as "12 heads in a row" ... a far, far, smaller probability than the 30% implied by the 70% forecast for the day.

How can I interpret the hourly forecast percentage to be consistent with the daily forecast?

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    $\begingroup$ I've wondered myself exactly how the events whose probabilities of occurrence are being provided are defined. That being said, you seem to be making an assumption of hourly rain occurrence probabilities as corresponding to independent occurrence of rain or not across consecutive hours in a day. That is likely far from valid for a variety of reasons, and incommensurate with how the forecast is made. $\endgroup$ – Mark L. Stone Apr 27 '16 at 23:51
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    $\begingroup$ As an extreme case, suppose there is a 50% chance it is going to rain tomorrow; and if it rains, it will be from 10 A.M. to 10 P.M. Then rain probability for the day is 50%, and rain probability for each hour from 10 A.M. to 10 P.M. would be 50%. $\endgroup$ – Mark L. Stone Apr 27 '16 at 23:58
  • $\begingroup$ @MarkL.Stone - Correct, I am pointing out that "making an assumption of hourly rain occurrence probabilities as corresponding to independent occurrence of rain or not" is obviously incorrect. What I'm struggling with is (a) a more precise definition of those hourly forecast percentages -- which make intuitive sense along with (b) how to convert a series of hourly forecasts to a daily one. $\endgroup$ – Steven L. Johnson Apr 28 '16 at 10:59
  • $\begingroup$ Looking further at my local NWS forecasts, it appears that the daily forecast % is equal to the maximum hourly percentage (rounded to nearest 10%). So, perhaps the interpretation is... if the conditions of this hour existed for 12 consecutive hours, there is a 50% probability of rain during one of those 12 hours. $\endgroup$ – Steven L. Johnson Apr 28 '16 at 11:03
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    $\begingroup$ How the weather varies from one hour to the next, or even one day to the next, is not like independent flips of a coin. You need a more sophisticated model to account for temporal correlation in conditions at nearby times. $\endgroup$ – whuber Aug 11 '16 at 20:56
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I was a little bit confused at first when thinking about the matter. However, the two main reason why you won't be able to harmonize forecasts on hourly and daily basis is the following.

The events for rain in breakdown perspective are not independent. Aggregating probabilities would require knowledge on how the events are related, but this will vary from case to case.

On my blog you'll find the article Rain Risk: Too much detail? which deals with the question and provides some neat images.

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  • $\begingroup$ Can you say more about how "The sampling on former weather data might be simply unfavorable" relates to interpreting hourly vs. daily percents? $\endgroup$ – Steven L. Johnson Aug 12 '16 at 21:12
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    $\begingroup$ In my article I pointed out that the simplest form of obtaining such probabilities consists of samling on historical weather data which arose under comparable circumstances (humidity, cloud formation, ...). My example shows that an unrepresentative sample can produce such a gap between daily and hourly probability. In practice the modelling process will be more complex, but still former data may play a role. I agree that this part of my answer appears confusing in the sole context of your question and I am going to remove it. $\endgroup$ – Jan Rothkegel Aug 13 '16 at 4:30

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