Conditional probability using dependent and independent tests

[anthonyk]

Hi, I'm trying to work out a probability value from a number of distributions (tests), it gets a little complicated because some of the tests are dependent on each other.

Lets say I have a surface which has black and white regions distributed over it, where black is "true" and white is "false". Test A covers a region of the space, and P(t | A) has a value of 0.5, meaning half of the space in A is black.

I have the following probabilities:

P(t | A) = 0.5
P(t | B) = 0.9
P(t | C) = 0.1

The regions overlap somewhat, according to the following probabilities:
P(A | B) = 1.0 (B is entirely within A)
P(B | A) = 0.1
P(C | A) = 0.05
P(A | C) = 0.5
P(B | C) = 0.3

Now i'd like to work out a meaningful measure of P(t | A and B and C) from the given information. If its important to have any other measures like P(C) etc let me know but I don't think its needed.

The value should be based on the values of P(t | A), P(t | B) etc, but should take into account the relative distributions of the regions. I'm pretty sure that the value P(t | A) should end up with no weight at all since region B is entirely contained within A and thus the other information about A is unimportant, but I don't know what maths will reflect this.

What this is trying to capture is there are a number of tests (A,B,C), and i'd like to give an overall estimate of the expected result from the given information, knowing that some of the tests are dependent.

If anyone can help me with this or point out the right topic to read up on i appreciate it. I'm not even sure if there is a meaningful way to combine the given distributions but if there is i'd like to hear it. thanks,
Anthonyps. I know you can say P(t | A and B and C) = P(t | B and C) since B = A and B, but if there is a way of handling incomplete overlapping regions it would be more useful.

 

[anthonyk]

This is roughly equivalent to someone having a number of tests to determine if they have a given disease. Test A, B and C all returned true. P(disease | A) = 0.5, P(disease | B) = 0.9, P(disease | C) = 0.1 (lets say C is a test that works out if you don't have the disease). I'd like to come up with an estimate of P(disease | AnBnC), if this is at all possible with the given information, or to identify what other information would be useful.