Example involving conditional probability and transitivity

In summary, the conversation discusses the application of the law of total probability for conditional probabilities and how to manipulate conditional probabilities using algebraic substitutions. The notation "P(T|A,F)" is not consistent with the notation used in the given image.
  • #1
hodor
7
0
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?

x0YhzYJ.png


Thanks
 
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  • #2
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
[tex] P(X=true) = P(X=true | Y = true)P(Y= true) + P(X = true | Y = false)P(Y = false) [/tex]
 
  • #3
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)?
 
  • #4
Ok, I found what I was looking for. It's an application of the law of total probability for conditional probabilities:

8UvY6P4.png
 
  • #5
hodor said:
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.
 
  • #6
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.
 

What is conditional probability?

Conditional probability is the likelihood of an event occurring given that another event has already occurred.

How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the intersection of two events by the probability of the first event.

What is transitivity in relation to conditional probability?

Transitivity is the principle that if event A is dependent on event B, and event B is dependent on event C, then event A is also dependent on event C.

How is transitivity used in conditional probability?

In conditional probability, transitivity is used to determine the likelihood of multiple events occurring in a sequence.

Can conditional probability and transitivity be applied in real-world situations?

Yes, conditional probability and transitivity can be applied in various fields such as finance, medicine, and weather forecasting to calculate the likelihood of certain events occurring based on previous events.

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