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)

 

[blue_raver22]

Hello Matt,

Thank you for your question. It seems like you are trying to understand how to calculate the daily chance of rain based on hourly probabilities. This is a common problem in weather forecasting and can be solved using dependent probabilities.

Firstly, it is important to understand that the chance of rain on a given day is dependent on the chance of rain in each hour. This means that the probability of rain in one hour will affect the probability of rain in the next hour. In other words, the hourly probabilities are not independent, but rather they are dependent on each other.

To calculate the daily chance of rain, you can use the formula P(A or B) = P(A) + P(B) - P(A and B). This means that the probability of it raining in any given hour during the day is equal to the sum of the probabilities of it raining in each hour, minus the probability of it raining in both hours at the same time.

For example, if the probability of rain in one hour is 0.1, then the probability of no rain in that hour is 0.9. Similarly, if the probability of rain in the next hour is also 0.1, then the probability of no rain in that hour is also 0.9.

To calculate the probability of it raining in both hours, you would multiply the probabilities together (0.1 x 0.1 = 0.01). Therefore, the probability of it raining in either one or both of those hours would be 0.1 + 0.1 - 0.01 = 0.19.

To calculate the daily chance of rain, you would apply this formula to all 24 hours of the day, taking into account the dependent probabilities. This can be a tedious process, but it is the most accurate way to calculate the daily chance of rain based on hourly probabilities.

I hope this explanation helps you better understand dependent probabilities and how to calculate the daily chance of rain. If you have any further questions, please don't hesitate to ask. Good luck with your research!

Best,