Chance of daily rain from hourly rain probability

[logistics86]

Hi everyone,

I was wondering if anyone could help me better understand dependent probabilities. I am interested in working out the daily chance of rain given the hourly chance's of rain.

Historically I know that on a given hour the chance of rain is 0.1. My first approach to work out the day chance of rain was:
P(rain during some time in the day) = 1-P(no rain during any hour)
= 1-(1-p)^24
= 0.9

However I know that the chance of rain on a given day is 0.4, significantly lower. I released hourly rain must be depended on other hours so. So for two hours I would do:
P(rain hour 1 or rain hour 2) = P(rain hour 1)+P(rain hour 2) - p(rain 1 and rain 2)
However doing this for 24 hours gets messy very fast.

Any ideas of how I could go about solving this sort of problem?

Thanks Matt

 

[Stephen Tashi]

Your calculations assume that the event of rain happening during one hour is independent of rain happening during another hour, which sounds unrealistic.

The two ways you attempted to solve the problem are equivalent. Completely working it out the "messy" way would amount to expanding the expression 1 -(1-p)^{24} symbolically before you substituted-in for p.

I think you need to find the precise interpretation of the two numbers that you have ( 0.1 probability of rain per hour and 0.4 probability of rain per day). Where did this data come from? Is there a document that defines how it was computed?

If this is a textbook problem, you should give the exact statement of it.

 

[logistics86]

Thanks for your replay Stephen. It's not a textbook problem, I'm just trying to work out a way that I can calculate the daily chance of rain given the hourly chances. The reason being a lot of websites such as
http://www.accuweather.com/us/ny/new-york/10017/forecast-accupop.asp?fday=1
Will give an hourly rain forecast and I'm interested in working out a day forecast from such an hourly forecast.

I have 120 days worth of hourly rain data from 20 nearby locations. From this data I calculated the hourly chance of rain as being 10%, i.e. it rained 1 out of 10 hours. The day chance of rain was about 40%, that is it rained on 48 out of 120 days.

Thanks Matt

 

[Stephen Tashi]

Investigate whether the probability of rain in hour n+1 really is independent of whether it rains in hour n.

(As a matter of terminology, what you are doing is not "calculating probabilities" since actual frequencies are not probabilities. You are "estimating probabilities".)

Group your hourly data into pairs of consecutive hours. Compare the fractions like:

(number of times in rained in the second hour)/ (number of pairs of hours)
vs
(number of times it rained in the second hour when it rained in the first hour)/ (number of pairs where it rained in the first hour)