Probability of getting through a slit In a double slit experiment with electron we cannot know which slit the electron went through. If we measure it then the interference pattern disappears. 
Is it possible to get a probability for which slit the electron went through given that it was detected at some place? For example if the electron was detected at the place of central maxima can we say that the probability that it came through slit A is p?
 A: Suppose you have some prior probability distribution for the state of the electron as it's emitted.  
Then, given the point where a given electron hits the detector, you can use Bayes's rule to update that probability distribution.
Now for any given initial state, you can calculate the probability that an observation at slit A will detect the electron.  Integrating over your Bayesian-updated probability distribution, you can calculate a number that could reasonably be interpreted as "The probability this particular electron would have been detected at slit A, had I chosen to make that observation."  
Of course if the initial probability distribution is concentrated on a single state (i.e. if you know the state of the electron as it's emitted), then the updating has no effect, so in that case the probability you calculate will be independent of where the electron hits the detector screen.  
A: A probability assigned to a retrodiction isn't science like the probability of a prediction.
And of you do it repeatedly you can't condition a current event on a future event. And if you recorded both so you could analyze them both later, then you would be working with the no interference population, so it would tell you zero, nothing, about how things worked when you didn't look.
If you tried to post select you'd sample from a different population, the population for a specific range of results on the screen, but then can't go back in time to measure which went through which slit. So the reproduction becomes a story, not a scientifically verifiable fact.
If you want to tell stories about what happens when you don't look then there are a range of stories. Some stories would say that it doesn't have a position when you sent measuring position. Others would say the positions can change in a completely discontinuous way when you aren't looking so it can go through both by jumping back and forth between the slits as it goes through. Others would insist that it has a position, well that the entire system has a configuration which is note and is essential. And those theories would still have some freedom, just like you can add a divergence free field to a current and not real change the dynamics of the the thing the current is a flow of. So for some currents it might be as simple as the leftmost portion of the screen is 100% due to the paths that went through the left slit and the rightmost portion of the screen is 100% due to the paths that went through the right slit and somewhere in between is one sharp dividing line related to the total flux through each it compared to the total flux on the screen.
But a different choice of current that differed by a divergence free field would model it differently (maybe the dividing line becomes a dividing swirl). And I would have said there would be by definition no way to tell the difference, but now I worry that maybe you can if you include all the back reaction you ignored to focus on just the frequency of the results.
But it still isn't a prediction about experimental results, it's just a story. No better than claiming that they couldn't have a position before you measured it. So the theory of quantum mechanics makes no predictions about it since it isn't a verifiable statement. And different interpretations will tell different stories.
A: The first experiments with electron diffraction behind an edge was carried out by Möllenstedt. He used a wire and get left and right the shadow of the wire intensity distributions.


Please take attention to the fact that not a double slit nor any slit was used. The next step of his experiments was to connect the wire with an electric source and to increase the electrical potential. The distribution pattern does changes. 

Why I tell you this? It is possible to answer your question

Is it possible to get a probability for which slit the electron went through given that it was detected at some place?

with YES for the cases, where the intenity distributions do not overlapping.
One has to suppose that electrons in such experiments are indivisible particles. But how then explain the distribution patterns? Electrons interact with the electrons on the surface of the wire and they build a common field. This field quantized. The flying electrons, dependent from their trajectory get deflected inside the common quantized field and hit the detector with a distribution in the form of fringes.
This explanation works very well for single electron experiments and for fringes behind a single edge too. But keep in mind that electrons as well as photons have wave characteristics due to their quantized field in any interaction.
A: Strictly speaking, it is not possible to compute a probability that an electron went through a given slit. However, it is possible to obtain a probability amplitude that the electron went through a particular slit, and this is one of the central tenets of quantum mechanics. 
Let us label the normalized vector in the Hilbert space associated with process of the electron going through slit A as $|\psi_{A}\rangle$ and the vector for the process of the electron going through slit B as $|\psi_{B}\rangle$. The resultant ket $|\psi\rangle$ for the process of the electron getting recorded at the detector in the absence of any possibility of knowing the which-slit information is the coherent superposition, 
${|\psi\rangle=c_{1}|\psi_{A}\rangle+c_{2}|\psi_{B}\rangle}$
where $c_{1}$ and $c_{2}$ are complex numbers which depend on the slit separation and the position of the detector. In fact $c_{1}=\langle\psi_{A}|\psi\rangle$ is the probability amplitude that the electron went through slit A and $c_{2}=\langle\psi_{B}|\psi\rangle$ is the probability amplitude that the electron went through slit B. 
However, the total electron count recorded at the detector is proportional to, 
${|\langle\psi|\psi\rangle|^2=|c_{1}|^2+|c_{2}|^2 + 2|c_{1}||c_{2}||\langle \psi_{A}|\psi_{B}\rangle|\cos\phi}$
and the third term is responsible for the interference fringes observed move our detector to change the variable $\phi$. Notice that this term is finite only when $\langle\psi_{A}|\psi_{B}\rangle\neq0$ i.e there is some indistinguishability between the two alternatives of propagation. 
It is incorrect to interpret $|c_{1}|^2$ as the probability that the electron took the path through slit A. This is because the electron never took any definite path in it's propagation. And it is precisely this fact, that the two possible paths of the electron were indistinguishable, that gave rise to interference in the first place. 
However, $|c_{1}|^2$ can be interpreted as the probability of detecting an electron if slit B was closed. Likewise the electron count recorded if slit A was closed would be proportional to $|c_{2}|^2$. But the existence of the third term makes it clear that the electron does not really take any particular path if both the slits are open.
