# Example involving conditional probability and transitivity

I'm just going to post a screenshot of the Example (free online textbook). I'm having a tough time making the leap to the first sum - what allows me to rewrite P(T|A) as the sum of the product of those two conditional probabilities? Thanks

## Answers and Replies

Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
It doesn't have anything to do with the fact that you have a conditional probability to start, it's an application of the more general statement
$$P(X=true) = P(X=true | Y = true)P(Y= true) + P(X = true | Y = false)P(Y = false)$$

Well I understand the bolded statement. I don't know why it doesn't have anything to do with the fact that I'm dealing with a conditional probability, since it's P( T = tr | A = tr ). In my mind I'm looking for an algebraic substitution or something that I know that allows me to manipulate this into something resembling P(T|A,F). What I don't know is where the P(T|A,F)*P(F|A) comes from or how I could get there.

For example, why not P(T|F,A)*P(F) and sum over F? Why is it P(F|A)?

Ok, I found what I was looking for. It's an application of the law of total probability for conditional probabilities: Stephen Tashi
Science Advisor
something that I know that allows me to manipulate this into something resembling P(T|A,F). What I don't know is where the P(T|A,F)*P(F|A) comes from or how I could get there.

What is the notation "P(T|A,F)" supposed to mean? It isn't consistent with the notation in the image you gave.

Just an attempt to shorthand what was in the image since I'm on my phone. That post should just be ignored at this point.