One of the thing that isn't obvious to me that seems to be for many people is the decision to maximize expected value instead of best worst case scenario. In this situation, given how exceptional the 100%1M vs. 50%50M situation is and how the 1M will definitely kill your financial problems, it really does seem like you'd like to pick the strategy that maximizes your worst case scenario (if choice=red, worst-case=1M; if choice=green, worst-case=0). I understand the reasoning behind expected values, I guess, it's just that it's not clear to me it is of any use here.
To me, the choice looks like "solve your financial issues with the red button; 100% chance" vs. "solve your financial issues and get extra money you won't really need, but with 50% chance through the green button".
I'd have a hard time choosing the green button.
It's curious because I'm a mathematician. I feel like I should know this better, but I've never really studied probability, much less statistics or economics.
(edit)
Another issue is what would it mean, in practice, that "50%" statement? I guess it means that if you'd play the game long enough, 50M would come out roughly half the times (by counting). This could mean a system in which the first 10 always fails, the second 10 always succeed, and the ones after that have their results based on a fair dice (1,2,3->50M; 4,5,6->0). This would certainly fit the frequency "definition". In practice, these probabilities don't mean a clean neat thing very often. Another issue is that the definition of that 50% means if you played that game long enough, you'd observe the half-half split, but you'll play it only once. Again, there is a statement about a limit (a statement about a_n, for n large), but you're only looking at a_1 (it often seems to me that people believe that information about EV transfers to information about a_1 -- it really does not). Even though I can mostly think of artificial examples (stuff like the one above), I'm not sure it'd be clear [in an actual situation] what is the meaning of that '50%'.
If the $1m "solves your financial problems" or is otherwise life-changing, you should almost certainly take the sure thing. As other discussions suggest, once you get into maybe the $3m-$5m net worth range, you presumably already don't have financial problems and another $1m is nice but not really transformative whereas $50m would be even though not a sure thing.
Even for a one time event, at some point it makes more sense to place the bet depending on a number of factors.
If it's hard to conceive of in this scenario, pick numbers about which it's easier to have intuition. What if you could take $10 for certain vs. a 50% chance of getting $500? Or pick some other values with the same ratio. 50% in this case just means a coin flip. You're right that no one gets the expected value. They get zero or they get $50m. But that may be a good bet depending on circumstances.
To me, the choice looks like "solve your financial issues with the red button; 100% chance" vs. "solve your financial issues and get extra money you won't really need, but with 50% chance through the green button".
I'd have a hard time choosing the green button.
It's curious because I'm a mathematician. I feel like I should know this better, but I've never really studied probability, much less statistics or economics.
(edit)
Another issue is what would it mean, in practice, that "50%" statement? I guess it means that if you'd play the game long enough, 50M would come out roughly half the times (by counting). This could mean a system in which the first 10 always fails, the second 10 always succeed, and the ones after that have their results based on a fair dice (1,2,3->50M; 4,5,6->0). This would certainly fit the frequency "definition". In practice, these probabilities don't mean a clean neat thing very often. Another issue is that the definition of that 50% means if you played that game long enough, you'd observe the half-half split, but you'll play it only once. Again, there is a statement about a limit (a statement about a_n, for n large), but you're only looking at a_1 (it often seems to me that people believe that information about EV transfers to information about a_1 -- it really does not). Even though I can mostly think of artificial examples (stuff like the one above), I'm not sure it'd be clear [in an actual situation] what is the meaning of that '50%'.