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I have a complex problem that I can simplify into what seems like an application of Bayes theorem. The information I do have is something like this:

I know it rains 60% of the time in this location (P(A)).

I know it rains 70% of the time if it is June (P(A | B)).

I know it rains 72% of the time if it is a Wednesday (P(A | C)).

What is the probability of rain on a Wednesday in June? Is that written the same as P(A | B ∩ C)? And using Bayes, this translates to:

P (A | B ∩ C) = P (A ∩ B ∩ C) / P (B ∩ C)

And basically from what I have read on this subject, I cannot really know this answer for sure because I dont know P(A ∩ B ∩ C). If I cannot answer this for sure, can I make a pretty close approximation by just averaging P(A | B) and P(A | C)?

I am referencing Conditional probability of event A given events B and C occur? Events B and C are independent from each other and How can I calculate the conditional probability of several events?

What I have so far is to make an average of the two measurements. So if I have sample sizes as follows:

P(A): 300

P(A | B): 25

P(A | C): 30

Then a good approximation would be P (A | B ∩ C) = (25*0.7+30*0.72)/(25+30)=0.71

Does this sound reasonable?

Marlowe
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  • Interesting question @Marlowe. Could you please clarify why you have added the word "Approximation" in the title of your question? – Dayne Nov 07 '19 at 07:12
  • This is my first time using the site so I am sorry if I put this in the wrong section. To give more context, I am working with a bet in the futures market and trying to apply the Kelly criterion. In order to use this I need a probability of winning the bet. I have quite a lot of statistics that I want to pump into generating this probability but it is not a requirement that I determine the absolute probability. If I can get something that is reasonably close that is OK with me. I thought it would make more sense to more people if I simplified the context. – Marlowe Nov 07 '19 at 16:39
  • @PeterFlom Can you take this off hold? This is a legitimate question and not homework – Marlowe Nov 11 '19 at 18:41

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