If a gambler and the casino keep going forever, how come the casino is always the winner?
In: Mathematics
The simplest example is a Roulette wheel. It has black, red, and two green squares. The chance of a person winning is only ever slightly less than 50%. Sure your gamblers will win sometimes, but over the long term, the house will win just enough to keep a stable income. Every casino game is designed this way. No matter how much they pay out, it will never be more than how much they collect from player losses.
Because the games they play are balanced in their favor.
Take roulette, for example. If you bet on a single number, the payout is 35-1. Bet $100, win $3,500. But there are actually 37 or 38 numbers on a roulette table, depending on location, because they’ll add a 0 and sometimes also a 00 to the wheel. So you aren’t going to win 1 out of every 36 bets, you’ll win 1 out of every 37 or 38. And that’s true for every other bet as well. Betting on a red or black number pays 1:1, but it’s not a 50/50 shot, because the 0’s are green and either bet will lose if one of those comes up. You can, of course, bet the 0’s if you want, but their odds follow the same pattern as well. The payout is less than the true odds, so given enough time, the casino will win on average.
Every casino game works the same way – if you compare the payout to the “true odds” of a particular spin of a wheel or roll of a dice, you’ll find that the payout is always less than the actual odds. There are only small exceptions – blackjack card counting works by finding a game with good rules (how many decks, how long between shuffles, how much a blackjack pays out, etc.) and increasing your bet when there are more “good cards” left in the shoe than bad cards. But even then, the odds are only slightly in the player’s favor, and they still have a chance of losing big on any given day, even if they might win over the long term.
An individual person might win in the short term, but the casinos know that whatever one person wins, they’ll make back from the dozens of other players lose. And, of course, it’s fairly likely that the person who wins will still keep playing and wind up losing the next time they play. They set the rules of the game, and they set them in their favor.
Intuitively, the house has to always win, otherwise it fails as a business. A business is there to make money – if the house always loses then it has to keep paying out and eventually goes bankrupt.
As for how the house always wins, it depends on the game. For example, roulette you can bet on black or red, but even if you place even bets on both, there’s also the green space(s) the ball can land on, so there’s always a chance that you lose on both bets.
Blackjack is one game where the house edge is very small, typically less than 1% using basic strategy. This has infamously led to things like the MIT card counting team which employed several strategies including card counting and team play to swing the odds to the player’s favor.
The odds are skewed in their favor. Take roulette for example. There are 38 numbers (37 in Europe) 1-36, 0, and 00 (00 does not exist in Europe)
If you bet equally on each number, they each pay 1:36, but there are 38 numbers. So for a $1 bet on each, you spend $38, and you win $36, so the casino made $2.
1st, 2nd, and 3rd 12 are also categories that each pay 1:3, so the house still wins on 0 and 00
Black and red pay 1:2, so the house wins on green (0 and 00)
Betting on green pays 1:18, it’s the same odds as betting on both 0 and 00, and there’s a 1/19 chance it lands on green, so the house still has the advantage.
Playing for short periods of time, you can beat the house, but the linger you play, the more likely it is, the house gets ahead, and the further ahead the house is, the less likely you are to get your money back. The house is constantly playing because they constantly have players in side, so they just keep getting further and further ahead.
Other games have different odds, and blackjack the game where the house has the least of an edge of .61%, as opposed to roulette’s 5.3% edge (2.7% for Europe)
Poker has no house edge, because you’re competing against other players rather than the house. The house makes money by either charging a fee for entry, buying drinks, etc
The game is always set up so your odds of winning are slightly less than “the house”‘s odds of winning.
A simple example: imagine there’s a coin toss game in the casino. They would set it up so you make a bet, the dealer matches it, the house keeps 1%, and if you win you get the 199% of your first bet. This sounds really good!
But it means your odds are not 50/50, they are 49/51. If you play this game long-term, you will always leave with less many than you started. You can leave while you are ahead, but if you keep playing you will lose. There are some betting strategies that can mitigate this, but 2 factors prevent that:
1. Casinos usually put a maximum bet at the table.
2. If you follow the betting strategy you very quickly need to be able to bet massive sums like $100,000 to stay ahead.
Every casino game is set up with a little “trick” like that that will make the casino be the winner in the long run. If the game allows players to place bets, there is almost always a rule in place to prevent strategies that always favor the player.
There are some cases like Blackjack where if a person is good at “card counting” and other strategies they can actually beat the house. Casinos are prepared for that, and if a dealer gets suspicious someone is counting cards they are asked to leave the table. If they move to another table and it happens again they are asked to leave the casino.
That’s the final line of defense: if someone starts winning “too much”, every casino intervenes. They either offer to put the person in a comped hotel room to entice them to keep gambling thus lose their winnings *or* they simply ban the person from the casino, assuming they’ve managed to somehow cheat without being detected.
If every game has at least a 50.1% chance of the player losing, then the casino is constantly making money overall no matter whether or not individual people win individual gambles.
Depends on the game. They’re specifically designed to be that way. There is no game where the player and the house do the exact same thing or play by the exact same rules.
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.
The house always wins because, from their point of view, they’re playing all the gamblers. Any particular gambler is only playing the house. In aggregate, the games are designed so that the house will make a profit from the games played against all the gamblers. If any one gambler wins some money, enough other gamblers lose more than that.
This is why you can have a small set of people who do win money from a casino, but the casino still “always wins”.
Another aspect of the “house always wins” is that, despite some transient gains, if you keep playing long enough, you will lose all your gains plus a little more.
Also not mentioned here, especially in places like Vegas, the entire town is basically a tourist trap where even if you do win, you’re likely going to spend some or all of that money there.
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