They are all misleading given that they just use the phrase “more money” and don’t specify the actual values.
Your reasoning should be:
A. If there is a combined total of $1,000 in the boxes then there are two possible outcomes:
- You win $1,000 (choose both)
- You win $0 (choose one)
B. If there is a combined total of $1,001,000 in the boxes then there are two possible outcomes:
- You win $1,001,000 (choose both)
- You win $1,000,000 (choose one)
It is true that A1 is greater than A2 and that B1 is greater than B2, but it’s also true that B2 is greater than A1. Given that we don’t know whether we are in A or B, and cannot force ourselves to be in one or the other, the comparison between A1 and B2 is relevant. This is why we must invert the above:
C. If you choose both then there are two possible outcomes:
- You win $1,000
- You win $1,001,000
D. If you choose one then there are two possible outcomes:
- You win $1,000,000
- You win $0
We can force ourselves into either C or D, and so don’t have to consider the comparison between C1 and D2 or between C2 and D1.
The only comparisons that matter are between C1 and C2 and between D1 and D2, and these comparisons must take into consideration the reliability of the predictor.
Given that the predictor is near-perfect, if I force myself into C then I will most likely win $1,000 and if I force myself into D then I will most likely win $1,000,000.