Conditional independence question - video 3.28

[Nancy Ibrahim]

Hello,

I don't understand how the professor solved the conditional independence quiz in video 3.28 and the similar homework question and I'd really appreciate any help with it. 

The question is about finding (T1=+ | T2=+) knowing that T1 and T2 are conditionally independent on C. I'm stuck at the very first step, i.e. how does he go from  (T1=+ | T2=+) to P(+2 | +1, C) P(C|+1) + P(+2 | +1, notC) P(notC|+1). He says we use total probability, but I don't see how we can add the C to make the second probability [P(+2 | +1, C) P(C|+1) + P(+2 | +1, notC) P(notC|+1)] equal to the first [(T1=+ | T2=+)].

Thanks alot,

Nancy

[Nancy Ibrahim]

Sorry the question is in video 3.23 - Conditional Independence II.
Thanks,
Nancy

[Anders Alm]

Hi,

I will try my best to explain.

We know that the first test is positive and that we are looking for the chances of the second test also being positive, but we do not know if cancer is present or not.

So we need to consider both options available:

Probabilities between independent (due to C being "known") events P(A and B) = P(A) * P(B):

What are the chances of T2 = +, given T1 = + and with cancer -> P(+2 | +1, C) * P(C|+1) -> P(+2|C) * P (C|+1)

What are the chances of T2 = +, given T1 = + and without cancer -> P(+2 | +1, notC) * P(notC|+1) -> P(+2|notC) * P (notC|+1)

Since these two options are mutually exclusive, P(A or B) -> P(A) + P(B), you just add them together.

I hope this was correct and that it helped :)

-Anders

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[Nancy Ibrahim]

Hi Anders,

Very helpful, I finally got it! :D

Thanks very much

Nancy

[lbjvg]

Great question and great answer.