Anyway, what inspired me to post this was this puzzle from the xkcd forums:
Source, with spoilersThe game is, the reaper will flip a biased coin. There is a 55% chance of heads, which will double your remaining earthly time, and a 45% chance of tails, which will reduce your remaining earthly time by 70%. The reaper will play this with you as often as you want, but you must state ahead of time how many games you will play. The games will be resolved instantly and exactly according to these rules with no cheating.
So, if you have 10 days left, and you play, you have a 55% chance of ending up with 20 days left, and a 45% chance of ending up with 3 days left. Doing the math, this has an expected value of 12.35 days. Awesome! Each game increases your expected remaining time by over 20%, and the most probable outcome is an increase in time remaining! What could go wrong?
So you declare you'll play the game with the reaper 1000 times. That's not living forever, but it ought to be pretty close! At first it was great, you got three heads in a row and you have months extra! But by the time you're done, your time left on this plane is best measured in Plank units! The reaper offers to reset the game, and play once again with 1000 tries if you feel like you were unlucky. You get even worse results! Indeed, this is a loosing game!
There are no tricks - for all x, the expected value is 1.235x (exactly). Even with multiple possibilities, the expected value for all of them increases, so overall EV increases (the 45% chance of 3 days gives another 45% of .9 days, 55% of 6, similar with the 55% chance of 20 days). However, if you run a simulation, you will indeed get a very small result.