Oh, I misbracketed your formula. Yes, 10^40.

I don't offhand have a model for why we expect your method to work, so I don't know why it fails. But another approach using the normal approximation gets within a factor of 10, so that shouldn't be it.

Um, I think you're just counting standard deviations in the wrong direction? You're counting standard deviations from 500,000 and doubling them, but the relevant distribution means are 510,000 and 490,000.

But no, those should be equivalent.

Oh! You're squaring a sum, not summing a square. You're counting the correct number of standard deviations in total, but you need the correct number for each distribution.

Dammit LW, stop nerd sniping me.