The large ratio is deliberate (and it's not so huge that 'all my theories are wrong!' is going to dominate).
The problem as stated is easy to misunderstand. I personally misunderstood (or "under-understood") it in at least three separate ways: 1. I considered the causal relation between Omega visiting me making that particular prediction and Alpha choosing me as potential receipant an unknown. 2. I considered what sort of predictions Omega would make in various counterfactuals an unknown. 3. I considered the truth value of "I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha's envelope." conditional on me alway...
This is a variant built on Gary Drescher's xor problem for timeless decision theory.
You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.
Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.
He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:
"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."
What to do?
EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.