We all know that it doesn't pay to gamble, but a week ago, I heard a paradox that seems to show the opposite- that you can earn money whenever you gamble. Let me show you how it works for a particular game- but after I'll show you it generalises to any. The game I heard it in the context of is Roulette.
Apparently there's a way to play the game where you just need to guess either black or red- both have an equal probability. You bet a certain amount of money and if you're right, they give you the amount you betted. If you lose, you pay them. Simple. So convince yourself that this is a game of pure chance, if you played this game a lot you wouldn't expect to win or lose anything. By the way, if you played this in a real casino the odds are slightly against you so that they make money- but that doesn't make a big difference to this paradox so let's look at the fair game for now. It seems like there is nothing you can do to improve the chance you make money. But let’s think about this strategy. What if you said, I'll always bet red. First time I'll bet 2. If I win, I'll stop. Otherwise I'll play again but this time I'll bet 4 this website. And if I lose again I'll bet 8, etc, etc. Then say you win on the 3rd round. You win 8, which covers all your previous loses. The beauty of this is, you will win eventually, and whenever that is, you win back everything you lost and then some. You might say, yeah but the amounts of money you're betting every time get huge quickly. And that's true. But it's pretty unlikely you'll even lose many times in a row. The probability you'll lose 3 rounds is 1 in 8. The idea behind this strategy is to chase your losses- you know, that thing that people tell you never to do. If you lost 3 rounds then you've lost 2+4+8=14. So you bet 2 higher than that total loss. That means that if you win this round, you get back everything you lose, and 2 more. You can check this yourself but this works for whatever round you finish up on, you bet your total losses so far plus 2. So whenever you play this game, as long as you go through until you eventually win a round, you always make 2. But this seems like a contradiction- you aren't supposed to consistently make money playing this game. You're supposed to break even. It gets worse than that, because you can apply this strategy to any betting situation, now matter the odds. You just bet a little bit more than all your total losses so far... But wait, before you bet on your house, you can see that this is clearly stupid for most betting situations. Most of the time, bets are stacked well against you. Let's say you're playing a game where you have a 1/10 chance of winning each bet -which is not at all unrealistic. Then the chance you'll lose even 12 games in a row is still very high, about 30%. But at that point you'll have to spend about $8000 and still are likely to lose more. So yeah, don't actually go do this in real life please. This strategy doesn't actually work- it's obvious why not when the odds are like this, but when the odds are 50/50, like in the roulette game, it's hard to see what the problem is. That's where the paradox is. I'm going to give you a week to see if you can figure it out. It took me a lot of frustrated hours with a friend until we cracked it, but it was good fun to think about which is why I wanted to tell you guys. It's a very interesting paradox. And if you do figure it out, here are some other variations you might try. What if you decide not to be too greedy, and you only bet your total loss plus 1 each time, instead of 2? Or what if you get super greedy, and you bet your loss plus some growing amount? In those two scenarios, see how quickly the amount you bet grows, and what your expected winnings are. If you need some hints, I have some small ones in the description. Good luck. Try it with a friend so you can discuss it- I highly recommend it. Oh and, promise me one thing- that you won't go gamble this week. Thank you.
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