and an example is provided for that:
Consider a simultaneous (independent) toss of two fair coins and a bell that rings whenever at least one of the coins lands on heads. Let the outcomes of the two coins be denoted X and Y, respectively, and let Z stand for the state of the bell, with Z = 1 representing ringing, and Z = 0 representing silence.
+ an extra condition is later added (page 43)
Suppose we do not hear the bell directly, but instead rely on a witness who is somewhat unreliable; whenever the bell does not ring, there is 50% chance that our witness will falsely report that it did. Letting W stand for the witness’s report
(full probability table provided at the bottom)
I understand that adding
Y="Heads" to the 2nd expression impacts the probability thus showing that the two variables X and Y are now dependent given W.
But, we also then get that
P(X = "Heads"| Y="Heads", W = 1) is equal to
P(X = "Heads"| Y="Heads") (=0.5) which is equation 1.2 of conditional independence... so how can it be? that the variables are both conditionally independent and dependent?
The probability table: