How do I combine probability estimates of two equivalent/mutually inclusive events? Let's say I'm pregnant with fraternal twins. One of them hangs out in the left side of my womb, and the other hangs out on the right (I have no idea how biology works). We've applied a flaky test to each fetus to determine its likely sex, and we know:
P(left twin is a girl) = 0.9
P(right twin is a girl) = 0.4
Let's assume these events are independent. What's the probability that I give birth to 2 girls? It's just 0.9 * 0.4 = 0.36, right?
Now let's go through the exact same scenario, but this time they're identical twins. Now what's the probability that I'm giving birth to 2 girls? These events are no longer independent. In fact, they're equivalent/mutually inclusive. Either both events occur, or neither event occurs.
Intuitively, I feel like I should start the same way as before...
0.9 * 0.4 = 0.36 # Some quantity proportional to the probability of having two girls
But also calculating
(1-0.9) * (1-0.4) = 0.06 # Some quantity proportional to the probability of having two boys
And because these are the only two possibilities, we can sum them to get the full probability mass, so the probability of me having two girls is
.36 / (.36 + .06) = 6/7 ~= .86
And the formula looks like:
P(A /\ B) = P(A) * P(B) / [ P(A) * P(B) + (1-P(A) * 1-P(B)]
Is my intuition right here? It seems to work out well for extreme values. If one probability is 1 and the other is 0, then the combined probability is undefined (makes sense). Otherwise, if one probability is 1, the combined probability is also 1, and same if one of them is 0.
And assuming this is right, is there a better way of formalizing it, or a name for the formula I'm applying?
Also, how do I reconcile this with this identity:
P(A /\ B) = P(A|B) * P(B)
Because, in the identical twin scenario, P(left_girl | right_girl) = P(right_girl | left_girl) = 1, this seems to lead to the contradictory conclusions that:
P(both girls) = 0.9
P(both girls) = 0.4
 A: In the identical twins scenario, the space of possible events is simply $BB$ or $GG$. Therefore, it does not make much sense to ask "what is the probability that one of the children has a particular sex" since their sexes are linked. In this case, you have actually made two measurements of one event $GG$, not one measurement for each of two events $Left-G$, $Right-G$. While it makes sense to take the probabilities at face value for the fraternal twin case, it breaks down when the results are actually conflicting.
The test results are saying that the probability of you having $GG$ is somewhere between $40\%$ and $90\%$. What you need to know is the precision of the probability estimated by each test in order to know what to do with the data. 
The probability model you would use is (where $(0.9,0.4)$ is the result vector):
$P(GG|(0.9,0.4))=\frac{P(GG)P((0.9,0.4)|GG)}{P(GG)P((0.9,0.4)|GG)+P(BB)P((0.9,0.4)|BB)}=\frac{P((0.9,0.4)|GG)}{P((0.9,0.4)|GG)+P((0.9,0.4)|BB)}$, assuming we are equally likely to have $GG$ or $BB$. Now, if the measurement error from each test is independent, then we can further simplify as:$P(GG|(0.9,0.4))=\frac{P(0.9)|GG)P(0.4|GG)}{P(0.9)|GG)P(0.4|GG)+P(0.9)|BB)P(0.4|BB)}$. What is missing from your specification is the model for the conditional distribution of the test results $T:P(T=t|GG)$ and $P(T=t|BB)$ for $t \in [0,1]$. 
For example, you could model it as a beta distribution, with the peak near $1$ for $T:P(T=t|GG)$, but substantial probability below $0.5$, and anther beta for $P(T=t|BB)$ but with its peak near $0$.
