If a gambler and the casino keep going forever, how come the casino is always the winner?
In: Mathematics
The games are designed so that on average players will lose more money than they win. As you play more games, the average of your outcomes approaches the expected average.
Say the casino has a game where you bet $10, flip a coin, and if you win the flip you get $19. This is close to how roulette works if you bet on red or black (instead of getting a <2x payout, you have a <50% chance to win due to there being slots on the roulette wheel where neither red nor black wins).
If you only play once, you have a 50% chance to win $9, and a 50% chance to lose $10. This means that on average, you’ll lose $.50 every time you play. There’s nothing preventing you from winning twice in a row and walking away having won $18… but you had a 25% chance of that happening.
The “house always wins” because they’re counting on the game being played a thousand times. Sure, there will be players who win a couple rounds in a row and walk away winners… but there will also be players who lose twice in a row out the outset and lose *more* than 50 cents per round they played.
The critical thing is that as more games are played, the range of outcomes gets closer and closer to 50/50. Winning 2 out of 2 games is a 25% chance. Winning 10 out of 10 games is a less than 1% chance. And because the house gains more each time a player loses than the house loses each time a player wins, the math quickly gets to the point where the odds of an outcome where the player have won *enough games beyond 50%* to beat the average 50 cent loss is near-impossible.
Extending the coin flip game above, players need to win ~53% of the time to break even, due to the $10 pay in and $9 payout. The more games are played, the less likely a 53% winrate becomes. 53 heads in 100 games is a ~30% chance. 530 heads in 1000 games is a ~3% chance. The more games are played, the less likely it is that the result stray far from 50/50… and the house edge makes it so that the results *do* have to be relatively far from 50/50 for the house to lose money.
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