You see two boxes and you can either take both boxes, or take only box B. Box A is transparent and contains $1000. Box B contains a visible number, say 1033. The Bank of Omega, which operates by very clear and transparent mechanisms, will pay you $1M if this number is prime, and $0 if it is composite. Omega is known to select prime numbers for Box B whenever Omega predicts that you will take only Box B; and conversely select composite numbers if Omega predicts that you will take both boxes. Omega has previously predicted correctly in 99.9% of cases.
Separately, the Numerical Lottery has randomly selected 1033 and is displaying this number on a screen nearby. The Lottery Bank, likewise operating by a clear known mechanism, will pay you $2 million if it has selected a composite number, and otherwise pay you $0. (This event will take place regardless of whether you take only B or both boxes, and both the Bank of Omega and the Lottery Bank will carry out their payment processes - you don't have to choose one game or the other.)
You previously played the game with Omega and the Numerical Lottery a few thousand times before you ran across this case where Omega's number and the Lottery number were the same, so this event is not suspicious.
Omega also knew the Lottery number before you saw it, and while making its prediction, and Omega likewise predicts correctly in 99.9% of the cases where the Lottery number happens to match Omega's number. (Omega's number is chosen independently of the lottery number, however.)
You have two minutes to make a decision, you don't have a calculator, and if you try to factor the number you will be run over by the trolley from the Ultimate Trolley Problem.
Do you take only box B, or both boxes?
I initially thought two-box, but on thinking about it more, I'm going for one-box.
For simple numbers, let's suppose that the lottery has a 50% chance of choosing a prime number, and that if Omega could select the same number as the lottery, he'll do so with 10% probability.
Three simple strategies:
1) Always one-box: Gets Omega's payout every time, wins the lottery 50% of the time. Average total payout $2M. (numbers are the same 10% of the time when the lottery is 'prime')
2) Always two-box: Omega never pays out, wins the lottery 50% of the time. Average total payout $1.001M. (numbers are the same 10% of the time when the lottery is 'composite')
3) Normally one-box, two-box when numbers are the same. Omega pays out 95% of the time. Lottery pays out 50% of the time. Average total payout $1.95M. (Numbers are the same 10% of the time when the lottery is 'composite')
The trick is that the question tries to lead you to the wrong counterfactual by drawing your attention to the situation where the numbers are the same. Whether you see the numbers being the same depends on your decision. In the counterfactual world where you decide something else, the lottery number doesn't change to match Omega's prediction. Instead, in the counterfactual world, the lottery number and Omega's number are different.
I'm not sure where you got that 95% number from for your strategy #3; it sounds like the "both numbers are the same" situation only happens once ever several thousand of runs.
Anyway, if you're using strategy 1, then if the two numbers are the same, that means that the number is prime, and your payout for this scenario is only 1 million dollars (you lost the lottery). If you're using strategy 3, then that means the number is not prime, and the payout is $2.001 million dollars (the number is not prime, because you're going to double box.)
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