> This experiments seems to make no attempt to eliminate
> the possibility that this is completely random.
You are correct, of course, that this doesn't prove that they see the photons. Maybe the experimental apparatus was poorly made and accidentally a noise when the photons were flashed. Or maybe someone cheated and told them which trials the photons would be delivered. Or perhaps they simply have ESP. But the one thing that we can easily have confidence about is when an effect differs from random chance.
"The subjects were asked to respond "yes" or "no" to say whether or not they thought they had seen a flash. The light was gradually reduced in intensity until the subjects could only guess the answer.
"They found that about 90 photons had to enter the eye for a 60% success rate in responding. Since only about 10% of photons arriving at the eye actually reach the retina, this means that about 9 photons were actually required at the receptors. Since the photons would have been spread over about 350 rods, the experimenters were able to conclude statistically that the rods must be responding to single photons, even if the subjects were not able to see such photons when they arrived too infrequently."
In other words, they lowered the intensity of the light until the subjects guesses were no better than random then concluded that 9 photons were spread over 350 rods.
The 1972 experiment assumed about 6 photons were entering the retina.
> "until the subjects guesses were no better than random"
Factor in that two photons should have a much higher chance of hitting neighboring rods which would substantially increase the chance of a hit (at the synaptic level) the question really should be "Why is it so much worse than random?"
I went back to the original paper [1] and the authors' conclusions are much more measured than précis we are all discussing. From page 838:
... the range of 54 to 148 quanta at the cornea
becomes an upper limit of 5 to 14 quanta actually
absorbed by the retinal rods. [ed: I presume this
is where the figure of 9 comes from]
3. This small number of quanta, in comparison to
the number of rods (500) involved, precludes any
significant two quantum absorptions per rod [ed: oops],
and means that in order to produce a visual effect,
one quantum must be absorbed by each of 5 to 14 rods
in the retina.
4. Because this number of individual events is so
small, it may be derived from an independent statistical
study of the relation between the intensity of a light
flash and the frequency with which it is seen. Such
experiments give values of 5 to 8 for the number of
critical events involved at the threshold of vision.
You may heat up a tea kettle a thousand times with a flame and conclude that the flame is heating the tea kettle, but perhaps every time the flame was too small and too far and it was in fact another source of energy hitting it from the environment warming it.
Our understanding of the world comes from substantial enough correlation that it moves into the realm of cause-and-effect. Though admittedly we often assume this too quickly.