I think I can help. You have set up a game of chance so that the expected value for the house (yourself) is negative. That means that on average you would have to pay out more than you would receive. However, while the payout is very big the chances of winning are very tiny so you wonder if this changes the game. In some sense, you are asking about the expected value of the game when you know the law of large numbers is not going to apply, because you are not going to play enough times for the ratio of wins to losses to average out.
This is a problem about sampling. The number of times you play the game will be much smaller than the number of games needed to yield the expected average. Suppose you conduct the game (only!) a million times. How reasonable is it to expect that you would collect a million dollars and not have to pay anything? In other words, we just need to calculate the probability of not having any “win” in the sample size of a million. The probability of a win in such a small sample size is tiny (epsilon) - so you wonder if you could consider it effectively zero and if it would be worthwhile to play the game.
The answer is that the chances are extremely high that you will not have to pay out anything (1-epsilon) so in almost every case it is lucrative to play the game. However, when you do lose, you lose so big that it (really does) cancel out the winnings you would be making in most case. So the expected value still holds—it’s not profitable to play the game.
My brain—and your brain too, probably—keeps buzzing that it is profitable to play the game because in almost every conceivable scenario, we can expect to make a million dollars. Human beings can’t correctly think intuitively about very small and very large numbers. Every time your brain buzzes on this problem—remind yourself it is because you’re not really weighing the enormity of the pay-off you’d have to pay. Your brain keeps saying the probability is small, but the product of the probability and the payout is a finite, non-zero number.
As several comments below have eluded, perhaps the impracticality of such a pay-off is detracting from the abstract understanding of the problem. However, this is a fascinating question, and should be addressed squarely. (I’m pretty certain you didn’t mean that you would just claim bankruptcy if you lost. Then your game would really be a scam, though I suppose we could argue about whether it is a scam in a sample where no one wins.)
I think I can help. You have set up a game of chance so that the expected value for the house (yourself) is negative. That means that on average you would have to pay out more than you would receive. However, while the payout is very big the chances of winning are very tiny so you wonder if this changes the game. In some sense, you are asking about the expected value of the game when you know the law of large numbers is not going to apply, because you are not going to play enough times for the ratio of wins to losses to average out.
This is a problem about sampling. The number of times you play the game will be much smaller than the number of games needed to yield the expected average. Suppose you conduct the game (only!) a million times. How reasonable is it to expect that you would collect a million dollars and not have to pay anything? In other words, we just need to calculate the probability of not having any “win” in the sample size of a million. The probability of a win in such a small sample size is tiny (epsilon) - so you wonder if you could consider it effectively zero and if it would be worthwhile to play the game.
The answer is that the chances are extremely high that you will not have to pay out anything (1-epsilon) so in almost every case it is lucrative to play the game. However, when you do lose, you lose so big that it (really does) cancel out the winnings you would be making in most case. So the expected value still holds—it’s not profitable to play the game.
My brain—and your brain too, probably—keeps buzzing that it is profitable to play the game because in almost every conceivable scenario, we can expect to make a million dollars. Human beings can’t correctly think intuitively about very small and very large numbers. Every time your brain buzzes on this problem—remind yourself it is because you’re not really weighing the enormity of the pay-off you’d have to pay. Your brain keeps saying the probability is small, but the product of the probability and the payout is a finite, non-zero number.
As several comments below have eluded, perhaps the impracticality of such a pay-off is detracting from the abstract understanding of the problem. However, this is a fascinating question, and should be addressed squarely. (I’m pretty certain you didn’t mean that you would just claim bankruptcy if you lost. Then your game would really be a scam, though I suppose we could argue about whether it is a scam in a sample where no one wins.)