If a gambler and the casino keep going forever, how come the casino is always the winner?
In: Mathematics
Like with any randomised event, be it gambling, dice throwing, coin tosses, the more you repeat the cycle the more consistently the probabilities are reinforced. If a coin toss landing on either side has 50% probability you could theoretically just get lucky and get 6 wins in a row by betting on tails. It’s not impossible. But if you do 1000 coin tosses in a row it’s very likely that the results will be more or less equal for either side. On every game in a casino the odds are stacked against the player from the get go and the casino hinges on gambler mentality, meaning they expect most people to play multiple times rather than just win once and leave. Quitting when you’re ahead is key in gambling. Even if you get lucky, if you keep going the casino is almost certainly getting its money back
Because the chances of the thing happening where the house wins on any given event is stacked in their favor.
Say you and me have 20 boxes in front of us. Under the boxes there are 12 black balls and 8 red balls that are randomly placed. Every time you pick a box with a red ball I give you a dollar. Every time you pick a box with a black ball you give me a dollar.
Over a long enough time, you’re going to pick black balls 60% of the time and red balls 40% of the time, because the physical laws of the game and randomness make it so.
If we play 1,000 times, you’ll take in $400 and give back $600 (I’m up $200)
If we play 10,000 times, you’ll take in $4,000 and give back $6,000 (I’m up $2,000)
.
.
.
If we play 100,000,000 times, you’ll take in $40,000,000 and give back $60,000,000 (I’m up $20,000,000)
And on and on and on
You never are the favorite, so you’ll never come out ahead in the long run
You pay me one dollar. You get to pick any 5 numbers between 1 and 11, including 1 and 11. I will use a random number generator to pick a number between 1 and 11, including 1 and 11. If the number is one of your numbers, I will pay you 2 dollars. If it isn’t, I keep your one dollar.
With 11 numbers able to be chosen, the monetary breakdown is:
6 x -$1 = -$6 (You can’t choose 6 of the 11 numbers, so you lose $1 each instance)
5 x $1 = $5 (You only win $1 with the 5 of 11 other numbers)
So every time you play this game, you will lose (-$6 + $5) / 11 = $0.09. $0.09 is a rather small number compared to the $1 bet that you placed for that one game. But if you play the game 100 times? Your expected value is -$9.
So by setting up every game to have payout odds so close to breakeven but not quite, you not only ensure that the casino will make money over the long haul, but that the player will win enough of the time to keep them coming.
If the odds of winning are 50%, and the prize for winning is to double your bet, then the more games that are played, the more the average earnings for the house and the players are going to break even.
Therefore, if the house wants to win, all they need to do is to either lower the chance of player victory or lower the payout. A 45% chance to win 1.9 times your bet doesn’t sound like much of a change, but when you run the numbers, it means that, instead of breaking even and earning a net $0 per game, every single game under the new odds gives the house an average of about 15% of the original bet. It doesn’t matter if more people win than expected for a while, the results will always trend towards the average which means that the house will slowly be building up money over time.
Most people don’t stop gambling when they’re winning. They think they can win more. And they lose. The expression “quit while you’re ahead “. Most gamblers can’t do that.
“Always wins” means they always have at least a 1% advantage in every game. Over the looooong run they can guarantee that they will always take in more money than they give out. Some people think “always wins” means they clean out every player. Nope.
They set the odds. Say you had a game based on a 100 sided die marked 1-99, if you wanted to ensure you make money you tell people if they win you will double their money, but they have to get a 51 or higher. In this case 51 or so percent of the time you the house wins, and 49 percent of the time the player wins.
Over a large enough amount of time you will win. You can better your odds by limiting the bet amount, this will make it so people that win big have to keep winning bug to double their money which is progressively unlikely since the odds are in your favor.
It’s all about what the casino nets at the end of the day. For every winner, there are hundreds of people that lose because of the nature of the games they are playing. Even having to pay out thousands to a winner, you’ve already made that much in just 5 or 10 minutes of other people playing and losing around the casino.
For games that don’t revolve completely around luck, like poker, the house just takes a portion of each winning pot at the end of the hand. This is called a “rake”.
So, the house always wins because at the end of the day, a casino is structured around games and their rules that will statistically always net them a profit… well, as long as people are playing the games anyway.
Imagine a dice with 3 sides.
If the dice lands on 1 or 2, the house wins.
If the dice lands on 3, the gambler wins.
Since the dice has an equal chance of landing on each side, the house will always win more than the gambler.
If I have a six sided die and offer to play you for a dollar, but the rules are that you win on a 6 and I win on anything else, if we play it 1000 times I’m statistically guaranteed to win overall. Pretty obvious, right?
Casinos operate on the same principle, but they can afford to make the odds a lot closer because of the number of people playing.
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