Probability of an event occurring given two conditions A seemingly basic question has been bugging me for a few months and no one I have talked to has been able to solve it. The problem is as follows:
m% of Harvard graduates are wealthy. n% of Virginians are wealthy. John is a Virginian and he is a Harvard graduate. What is the probability P that John is wealthy.
A few observations:
1) It has been suggested to me by several people to simply "OR" the two conditions together, in other words treat the two conditions as two "dice rolls". I.e. P = 1-(1-m)(1-n). To see why this does not work, let n = 0. This makes P = m. This is wrong because if n is zero, then no Virginians are wealthy and since John is a Virginian this means that P should be 0, not m.
2) If in addition to n% of Virginians being wealthy, n% of people in the world take as a whole are also wealthy, then n% of Virginians being wealthy is not useful information, so P = m. "m% of Harvard graduates are wealthy" means that m% of Harvard graduates in the world are wealthy, so if the percentage of wealthy people in Virginia is the same as the percentage of wealthy people in the world, then the percentage of Harvard graduates that are wealthy in the world should not differ from the percentage of Harvard graduates that are wealthy in Virginia.
You might say based on (2) that there is not information, but I think you are given two important pieces of information which you should be able to use to make a prediction that is more informed than someone who is not given this information.
 A: Greetings from Reddit! (Currently not a user.) I was perusing through it and found your post. lulu actually already answered your question in their comments, but I can present the answer more clearly here. (P.S. I'm a programmer, not a mathematician.)
The answer is that you can't solve this problem with just the given information; that is, there is a lack of sufficient information to solve it.
I'll illustrate why this is with some rudimentary pictures I drew in Paint. (Note: ignore the actual proportions I use in these diagrams, they're irrelevant to the point I'm trying to make.) First, we have a Venn diagram of Harvard graduates and Virginians, and we know that John is both - that is, the intersection of both circles:

Now, your problem states:

m% of Harvard graduates are wealthy. n% of Virginians are wealthy.
  John is a Virginian and he is a Harvard graduate. What is the
  probability P that John is wealthy.

What you were probably thinking was, "Since we know the percentage of wealthiness of each group individually, there must be some way to combine these two to get the likelihood that John is wealthy," and while that is technically true - this could be determined with more data - this thinking is however incorrect given only the information supplied by the problem.
Here's why. Let's draw the percentage of wealthiness in the Harvard graduate and Virginian groups and see where they overlap:

Err... well... wait a minute here... We know the percentages, but where do we draw the wealthiness? Maybe it's actually like this:

or perhaps like this:

And so, as you can see, without first knowing the relationships between the 4 groups:


*

*Wealthy Harvard graduates

*Non-wealthy Harvard graduates

*Wealthy Virginians

*Non-wealthy Virginians


There's no possible way to rightly solve the problem as stated, and in fact, the percentage likelihood that John is wealthy, as far as we know, could range anywhere from 0-100%!

Also, here is the information lacking in the problem... You would need to determine these two things:


*

*What percentage of Wealthy Harvard graduates (#1) (or Non-Wealthy Harvard Graduates (#2), since #1 + #2 == 100%) are Virginians (#3 + #4)?

*What percentage of Wealthy Virginians (#3) (or Non-Wealthy Virginians (#4), since #3 + #4 == 100%) are Harvard graduates (#1 + #2)?


With the above information, you would then know where to draw the wealthiness in the Harvard graduates and Virginians groups, and you could find your answer within the intersection of the two groups (green), wherever there is wealthiness within said intersection.

Another, more complicated note: since wealthiness is wealthiness, you would have to make sure that once you know the further information, you draw your Venn diagram properly... One of the graphs I drew above would actually be invalid with this new information:

This is because the Harvard graduates circle's wealthiness is only some 80% or something of the intersection, whereas the Virginians circle's wealthiness is consuming 100% of the circle... You can't say that both 80% and 100% of the intersection are wealthy... If 100% of them are wealthy, the Harvard graduates circle's wealthiness portion would have to be rotated over such that it covered 100% of the intersection as well. Otherwise, your newly provided information is incorrect/conflicting/contradictory. :P
