Duncan Smith wrote: > Sticker wrote: > > I have a question, how to calculate the probability as following: > > P(A1 & A2 & A3 | B)? > > Given the condition that event B happens, how to calculate the > > probability of all A1, A2 and A3 happen? Here A1, A2, A3 are > > independent with each other but they are dependent with B separately. > > > > Can I say: > > > > P(A1 & A2 & A3 | B) = P(A1 | B) * P(A2 | B) * P(A3 | B)? > > > > That's OK if the As are conditionally independent given B, but you seem > to be saying they are not. > > Duncan
As Duncan said it is not clear that what is the relationship among As + B., let ma give an example about conditional independence.
Suppose John and Jan each toss a coin I give(EVENT B) separately. Let A1 represent John's toss outcome, and A2 represent Jan's toss outcome. It would be reasonalbe assuming that A1 and A2 are independent for a coin I give(EVENT B). However A1 is related to B and so is A2 in a sence if a coin I give is biased toward head/tail.