Sorry for my poor phrasing. The Number Lottery’s number is randomly chosen and has nothing to do with Omega’s prediction of you as a two-boxer or one-boxer. It is only Omega’s choice of number that depends on whether it believes you are a one-boxer or two-boxer. Does this clear it up?

Note that there is a caveat: if your strategy for deciding to one-box or two-box depends on the outcome of the Number Lottery, then Omega’s choice of number and the Lottery’s choice of number are no longer independent.

Of course! I meant to say that Richard’s line of thought was mistaken because it didn’t take into account the (default) independence of Omega’s choice of number and the Number Lottery’s choice of number. Suggesting that there are only two possible strategies for approaching this problem was a consequence of my poor wording.