One good example of how statistics comes into play in quantum mechanics is with the famous dual-slit experiment that we've all done in high-school physics.
I remember Mr. Van Orden trying to set up the experiment for us. The primary challenge for us was creating the dual slits through which to shine the red light. The idea was to hold two razor blades together to scribe two very fine and closely-spaced lines through the carbon on a soot-blackened microscope slide. As I recall (and you know how poor recollections can be after so many years), he finally offered an A grade to anyone who could succeed in making the slits on the slide. Well, that was motivation enough for us, and I was the fortunate one who finally succeeded.
Once we had the slide, we shined a red light through the slide, and observed an interference pattern created by the interaction of light waves after passing through the two closely-spaced slits.
This experiment can also be duplicated using electrons instead of photons.
The curious thing is that if you send one electron or photon at a time through the the apparatus, you still get an interference pattern that builds up over time. That doesn't seem right, since photons and electrons are so small that they must go through either one slit or the other, not both. The accumulated result over time should be two bright lines on the detector (photographic film, for example).
One of the theories proposed to explain this is that the entity (photon or electron) takes all possible paths through space between the emitter and the detector, and where it shows up on the detector is based on the statistical probabilities of the infinite paths it has simultaneously taken.
To the average man on the street, this is nonsense. Same thing for women and children.
What I think is really happening is that the photon or electron grows in size when it leaves the emitter. Think of it as a spherical blob of jello. If there is nothing between the emitter and the detector, the blob is consumed by a single molecule of the detector upon first contact with the detector, which will be on the straight line between the emitter and the detector. Think of a tennis ball hitting the wall, and which piece of fuzz on the ball makes first contact with the wall.
The blob cannot be detected by more than one molecule of the detector, because the act of detection requires the whole blob--this is a basic limitation of measurement. So the first detector molecule contacted by the blob, being thirsty for jello, sucks in the whole blob and you end up with a tiny point on the detector.
Now, send the blob through the two slits. Think of the blob squeezing through both slits. It doesn't need to ever really separate into two pieces, because the front part can rejoin on the back side before the back of the blob has gone through. Or perhaps the blob is a five-dimensional blob (more on this some other time), and it may appear to split in our four dimensions but really doesn't. Anyway, the act of passing through the slits causes it to become misshapen have strange undulations going on.
When it hits the detector, the first part of the blob to hit the detector, as before, causes the whole blob to be sucked into a single detector molecule, again making a point on the detector. But now the shape of the blob is strange. In fact, the shape of the blob is probably pretty much the same as the interference pattern that results after many blobs have been through the two slits.
If we could know precisely the shape of the blob at every instant in time, we could with precision tell where each photon or electron would strike the detector. But since we can't know that--it's far too small--we use statistics to establish the probability that it will hit the detector at any given point. So, we are using statistics to desribe the action of something that is too small to measure. It isn't the case that the reality at really small scales is a basket full of statistical probabilities.
