I’m reading The Fabric of the Cosmos right now, and there’s something in it that puzzles me. I think Greene may have made a mistake in one of his analogies. Either that or–more likely–I’m not getting something.

Anyway the analogy is an attempt to explain Aspect’s experiment proving quantum non-locality. I’m really interested in the analogy itself, though. I’m aware calculating quantum mechanical probabilities is different than calculating “conventional” probability. For those of you with access to the book, this is from pp. 107-110 of the Vintage paperback.

Anyway, the story is Mulder and Scully (Greene loves his pop-culture references.) are on vacation in widely seperate locales. Both receive a mysterious package containing hundereds of small titanium cubes. The cubes can be opened three ways we’ll call top, right and front. Inside the cube is a sphere which glows red or blue at random. It will glow a different color depending on what door is opened first.

Mulder calls Scully and claims that the packages have been sent by aliens. The aliens have a mysterious tech that works like this: If Mulder opens, say, the front, and it glows red, then the device will communicate a signal to Scully’s Cube making it glow red if she opens up the front. If she opens up the right or top, the Sphere will flash at random. Scully thinks this is dumb. The boxes were simply programmed from the outset to flash the same colors if the same doors are opened. Since there is no way to test for either hypothesis–the flashing is disabled if the boxes are tampered with–it’s undecidable.

Mulder puzzles on this for a while, then comes up with an idea. They will open up the boxes in order (they are labled) but open the doors at random. If Scully is correct then they should see the same color more than 50% of the time.

For example, say the box is programmed to flash RED(top) RED(right) BLUE(front). There are nine possible ways M and S can open the doors. TT, TR, TF, RT, etc. etc. Five of those ways give the same color: TT, TR, RT, RR, FF. The same reasoning would hold for any cube which might flash both colors, since they must have two doors flashing one color and one door flashing the other. Thus M and S must see the same color *at least* 5/9ths of the time. If and cube is either RRR or BBB they will see the same color always, so this will only increase the percentage of same color sightings above 50%.

So far so good. Mulder has shown that if Scully is right and the boxes are pre-programmed they will see the same color more than 50% of the time.

But my problem is that if Mulder is right and the boxes are not preprogrammed, they will *still* see the same color more than 50% of the time.

Say Mulder opens the front door and it glows red. Now Scully has a 1/3 chance of opening the front door, in which case she will also see red. Scully has a 2/3 chance of opening a different door, in which case the sphere flashes at random, giving her a 50-50 chance of seeing red or blue. So Scully has a total chance of 2/3 of seeing red. Thus she will see the same color as Mulder 2/3 of the time.

Granted, 2/3 is a different number than 5/9. But under Scully’s hypthesis, enough cubes could have been programmed RRR or BBB to raise the percentage of same color sightings to 2/3. In short the experiment cannot decide whether Mulder or Scully are correct.

Why am I wrong?