Given three random real numbers A, B, and K, what's the probability that K is between A and B? Does it matter what probability distribution you use to choose K? If so, shouldn't there be a "best" K, e.g. K = 0, to choose each time?

A few things, when you say you choose A, B, and K "randomly". You are not actually telling us enough information. On what range are you choosing them? Are they real numbers, rational numbers, integers, natural numbers? Also, just saying I choose an integer at random doesn't make any sense for multiple reasons. To illustrate the main one, say I tell you I want to pick a number from 1 to 10 at random. Well, I can do that MANY ways.

I can flip a coin and pick 1 if tails and 10 if heads. (so 1 has probability 1/2, and 10 has probability 1/2, but every other number has a zero probability). I could devise many ways to do this, roll a die and flip a coin and label those combination to fit some value from 1-10. Now, you may ask, why not choose each one evenly?

Ok, so I choose 1 with probability 1/10, 2 with 1/10, etc. That's fine, but now the range of numbers I look at is VERY important. If i change to being from 1-100, now each number has a probability of 1/100. Now, if you want to do this for all positive integers what is the probability? Well, before it was 1/(number of things), but now it is 1/infty? Then everything has a probability of zero. That's a problem. Now, I won't choose anything...

Now, I can answer your question for A, B random, and K random, what is the probability of A<K<B? Well, this is a problem in statistics and is known as Order statistics, where you are choosing n=3, and looking at the 2nd order statistic. Now, calculating that probability I need to be given how you are choosing the numbers, once you give me that, then I could try to answer your question. (and at the moment I'm blanking on how to correctly answer that).

/r/math Thread