I was browsing a well known technology board and someone posted a pretty standard question about conditional probabilities. As this happens to be the current subject of my lectures (just starting), I decided to take a swing at them and it turns out I have no idea what I'm doing, or what kind of proof (contradiction?) I should be using in these kinds of questions. I can show these are correct/incorrect with arbitrary numbers, but I have a gut feeling there are more succinct ways of showing truth.
If P(a|b,c) = P(a), then P(b|c) = P(b)
I get the feeling that this one is wrong, only because we can't assume c has no bearing on b with only the information that neither affects a. From my limited knowledge of conditional probability, I'm reading the first statement that a is independent from both b and c.
I did this, where I assumed the implication was true, and got a contradiction. Nevermind, I made the assumption that P(a|b,c) = P(a) ==> P(a|b) = P(a) and P(a|c) = P(a), and I apparently can't do that?
If P(a | b) = P(a), then P(a | b, c) = P(a | c)
I believe this one is true, because if a and b are independent, then the probability of a, given two conditions - one of which doesn't affect it, and one that might - should just be the probability of a given the other condition (c). I also tried proving this one, but I kept going in a circle.
I've tried contradiction approaches, but I'm not familiar enough with conditional probability to say something is for sure a contradiction (For example, before I realized the incorrect assumption I made on the first one, I assumed the second part was true and arrived at P(b|a,c) = P(b|c), which looks false, but because of the stuff about independence I can't be certain). Most of my 'technique' has revolved around using the definitions
P(a|b) = P(a,b)/p(b) (expanded to) P(a|b,c) = P(a,b,c)/P(b,c)
P(a,b) = P(a|b)P(b) = P(b|a)P(a)
Any help is appreciated, especially if you focus on the "why" of choosing a certain proof.