Can someone help me out here? I have a hard time grasping this concept. I’ve learned this before but I had a hard time understand it then.
I don’t get Reading 8 at all. I don’t understand P(A|B) or P(B|A) and how I can use Bayes’ formula to convert between the two. I don’t understand why the joint probability is also P(AB) = P(A|B) * P(B) and not just P(AB) = P(A) * P(B).
Can someone explain to me thoroughly or can direct me to videos clearly explaining this? I’ve gone through the CFAI, Elan videos and Schweser videos.
It’s hard to wrap my head around P(A|B) and P(B|A). I don’t want to just memorize the formulas because I know I’ll forget without thinking of it logically and intuitively.
These two are actually the same.
P(AB) = P(A|B) * P(B)
P(AB) = P(A) * P(B).
Think of it this way. Let’s say the probability that your computer will crash is P(A) and the probability that a republican will become US president is P(B).
What are the chances the crashing of your computer will have a direct effect on whether or not a republican becomes president?..of course, there’s no relationship between both. If your computer crashes, it wouldn’t affect who becomes the president…and conversely who becomes the president will have absolutely no effect on your computer crashing.
So if your computer will crash with a 10 percent probability, this event will happen with a 10 percent probability regardless of who becomes the president.
So, P(A|B) = P(A)
Now on to another example.
Let’s say Probability that you will stink = P(A) and Probability that you will not shower = P(B)
Pr(You will stink if you don’t shower) = Pr(You will stink) * Pr( You will not shower)
But oftentimes, you can only stink only if you haven’t showered. So unlike in the previous example, there’s actually a direct relationship between stinking and not having shower.
So Pr(you will stink) will only happen (given you haven’t showered)
So P(A) = P(A|B)
I hope this hasn’t confused you further.
This seems to give a very clear derivation of the bayes formula, using diagrams to help visualise the theory.
Regarding the question of independent and dependent events as copied from some website:
Multiplication Rule 1: When two events, A and B, are independent, the probability of both occurring is: P(A and B) = P(A) · P(B)
Multiplication Rule 2: When two events, A and B, are dependent, the probability of both occurring is: P(A and B) = P(A) · P(B|A)
WOW!. Thank you. Thinking about Bayes’ Theorem like a venn diagram is sooo helpful.