Physicist's Double Slit Experiment: How is it Possible?

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I have read that physicist have been able to do the double slit experiment in which they were able to release one photon or one electron at a time. How is this possible? How would a person know that only one photon or one electron had been released? And how is this done?
 
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(where did you read this? Why do always people forget to tell this important information)

One simple adjust the beam so that without the slits a desirable hit frequency is obtained on a big screen. Then one installs the slits, and hope that the same frequency is still there
 
When I did this in a course a couple years back with a double slit and a lightbulb, we just used some very ... "powerful" filters to decrease the intensity a tremendous amount. Then we proceeded to use more then 1 :).
 
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
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