@TomHsiung wrote: I think the last two formula means A = 1 and B = 0 as well as A = 0 and B = 1 are independent, respectively. That's why I'm confused about the independence between event A and B.
Hello @TomHsiung,
I see your point.
My understanding is that the random variables describing the individual responses are independent, hence the products of probabilities in the Wikipedia article you have mentioned. Yet, as a rule, matched pairs are correlated in the following sense: If X, Y denote the responses of a randomly selected matched pair, these random variables X and Y are usually dependent because they tend to have similar response probabilities due to the matching.
Here is a small example with only two matched pairs: (X 11 , X 12 ) and (X 21 , X 22 ), each representing, say, (case, control).
Assume independent Bernoulli distributions X 11 ~B(1, 0.94), X 12 ~B(1, 0.91), X 21 ~B(1, 0.22), X 22 ~B(1, 0.29).
Define X:=X U1 , Y:=X U2 with a random variable U, independent of the X ij , describing the selection of a pair: P(U=1)=P(U=2)=0.5.
Then we obtain the joint distribution of (X, Y) from calculations like P(X=1, Y=1) = P(U=1)P(X 11 =1)P(X 12 =1)+P(U=2)P(X 21 =1)P(X 22 =1)=0.4596:
P
X=0
X=1
Y=0
0.2796
0.1204
0.4
Y=1
0.1404
0.4596
0.6
0.42
0.58
1
Now we see that X and Y are correlated, hence dependent: Their correlation coefficient is r X,Y =(0.4596-0.58*0.6)/sqrt(0.58*0.42*0.6*0.4)=0.46155...
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