Assume Y is caused by two independent events A & B, upon investigating a data set carrying 1000 entries we see.
$$\begin{align}\text{Number of occurrence of events } A = 497 \text{ and } B=472.\\ \text{ Out of 497 occurrences of A, A causing Y to happen = 259 (A')and }\\ \text{ out of 472 occurrences of B, B causing Y to happen= 249 (B')} \end{align}$$
The observations says either A or B can cause Y and the number of times Y has happened = 381. Also joint distribution of $$\begin{align}Y|A \cap B = 127 \end{align}$$
So, If I rewrite the whole thing, this is what I see,
$$\begin{array}{lrr}\\ p(A) =0.497 \\ p(B) =0.472 \\ p(Y|A) =0.259\\ p(Y|B) =0.249\\ p(Y|A\cup B) =0.381 \text{ - A or B can cause Y } \\ p(Y|A\cap B) =0.127 \text{ - A and B can cause Y } \\ \text{ Now, }p(Y) = p(Y|A)+p(Y|B) - p(Y|A\cap B) =0.259+0.249 - 0.127 = 0.381 \end{array}$$
In case if we don't have the joint probability distribution $p(Y|A\cap B)$ =0.127, is there a way to find p(Y) = 0.381?
Thanks,
Kamal.