Setting up Conditional Probabilty formula for November temps being < t

Hello
(first time to post)

I have a question on setting up a conditional probability formula. At lest I think it falls under conditional probability.

I am trying to assign (calculate) a probability that the low temperature (for November in this case but any month going forward) will be less than "t"

Here is what data I have: going back to 2009 I have the lows for each November. from that data ALONE (treating the 9 points as normally distributed) I can say 98% chance the low will be less than 30 degrees and so forth (in Excel)

My conditional question comes in here:
The probability of a November low being less than 18 fromthe historic lows is 15%

What if I see the weather forecast shows a forecasted low of17?

Now I have a “given the forecast says..”

Next, every morning I gather and store in a database: theforecast and actual temps (yesterday). Meaning I can with SQL scripts gather the probability that the forecast is accurate within “x” degrees
I do this for two different forecasts services. I have plenty of data.

I know that the forecast needs to be considered. In June ifI was asked (for planning purposes) what is the probability November’s low will be less than 18, I would need to say 15% (from history)

But if I am asked now and I see the forecast shows 17degrees in 3 days, I need to include “given that the 3 day out forecast states 17 and the accuracy of being within 1 degree 3 days out is 55% (made that up)

How can I use the data I have: historical and within my databases to answer the question: what is the probability the temperature willbe lower than “t” for the month?
Thanks
-Sparky_

There may be more sophisticated answers, but I would forget the 9 data points on historic lows for two reasons.

First, it is a tiny data set on which to base conclusions.

Second, tomorrow's weather is not a true random variable. (Poincare has a neat essay on why probability theory works for things that are fundamentally deterministic.) All sorts of things affect such tomorrow's temperature, and weather forecasters take as many of them into account as they can.

So I would use the much larger data set to determine what is the distribution of errors between actual lows and forecasted lows in all the Novembers for which you have data. Then you can make a much more reliable estimate of the probability that tomorrow's low will be below 18 given a forecast of 17.

If you don’t mind, I would like to discuss a little further, what I’m needing to answer is “whet is the probability “T” will be the absolute low for the month.

Using the small data set which contains the absolute lows (for the last 9 years for the specific month) and fit a distribution suited for extremes and small data set I can answer the generic question. (this step seems straightforward)
A second somewhat straightforward step is to answer the probability of the forecast being “accurate”. I can easily report forecast accuracy from my database – very straightforward if I can get the SQL scripts correct.

I am stubbornly stuck (not arguing – I admit I don’t have a handle on this) with the mindset that these two should be used to answer the question, “what is the probability “t” will be the low for the month” given the known forecast and historic lows. I sure want to setup a conditional probability equation or use Bayes. I’ve read through some conditional probability examples (from my stats book – been many years since I’ve I had it) and they seem close to this but not quite the same problem.
Can we discuss a little further?

Thanks so much!
-Sparky_

If you don’t mind, I would like to discuss a little further, what I’m needing to answer is “whet is the probability “T” will be the absolute low for the month.

Using the small data set which contains the absolute lows (for the last 9 years for the specific month) and fit a distribution suited for extremes and small data set I can answer the generic question. (this step seems straightforward)
A second somewhat straightforward step is to answer the probability of the forecast being “accurate”. I can easily report forecast accuracy from my database – very straightforward if I can get the SQL scripts correct.

I am stubbornly stuck (not arguing – I admit I don’t have a handle on this) with the mindset that these two should be used to answer the question, “what is the probability “t” will be the low for the month” given the known forecast and historic lows. I sure want to setup a conditional probability equation or use Bayes. I’ve read through some conditional probability examples (from my stats book – been many years since I’ve I had it) and they seem close to this but not quite the same problem.
Can we discuss a little further?

Thanks so much!
-Sparky_

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