字幕表 動画を再生する 英語字幕をプリント This video was made possible by Brilliant. Learn more about the math and science behind this video’s topic for 20% off by being one of the first 200 to go to Brilliant.org/Wendover. Within the walls of the 51 licensed casinos of the Las Vegas Strip, there are 2,879 gaming tables collectively bringing in $3.1 billion in revenue annually, or over a million dollars each. In addition, there are some 38,864 slot machines bringing in another $3.4 billion. A single large Las Vegas casino, like the Bellagio, make more annually than some small countries. This is because the Casino business model is pretty much bulletproof. Overall, the odds are always in the Casino’s favor—if this weren’t true, the casino would fail—so these floors just print money. MGM International, for example, one of the world’s largest multinational gaming companies, has about 2.5 million square feet of Casino floor worldwide meaning it makes, on average, $1,138 per square foot. With the money their casinos bring in, they could line every inch of every foot of their casino floor worldwide with a brand new iPad and still have money left over. This is all to say, casinos make a lot of money, but to do so, they need a lot more money coming through their doors. You see, the casino business model all revolves around risk. With every game they have, the odds are in their favor, but that’s not to say the house will always win. Their advantage varies from game to game—in roulette, it’s about 5.25%, in Poker, it’s about 3.35%, and in Blackjack, it’s about 0.5%. Of course, given how tight these margins are, there's a natural variability so casinos can come out behind on a given table on a given night, but overall, with enough tables and enough nights, they’ll average out to these odds. However, in order to do so, they need an immense amount of money running through their casinos. Blackjack, for example, is one of the games with the lowest house edge, so if a casino wants to earn $1 billion in a given year from the game, which would not be an unreasonable estimate for a large gaming company like MGM International, they would need $200 billion changing hands within their doors each year. $200 billion is an enormous amount of money. That’s pretty much the entire GDP of New Zealand, passing through, in the case of MGM, a physical structure barely larger than the Empire State Building. When you have such a rapid throughput of money, very slight changes in the odds can make a huge dent in the gaming company’s earnings. If, for example, the house edge in Blackjack changed from 0.5% to 0.4%, they would lose $200 million, assuming $200 billion in annual play. This is why making sure these odds stay in their favor is so important to casinos. It can quite literally make or break them. Despite what pop culture might portray, a casino’s biggest problem is not robbers or hackers or even technically cheaters, because each of those is relatively easy to prevent. Rather, their biggest problem is people who are able to turn the odds in their favor without robbing, hacking, or even cheating. You see, in most common-law countries, such as the US, England, Ireland, or Australia, cheating is legally defined as altering the outcome of the game, acquiring knowledge not available to all players, or changing ones bet after learning of the outcome. Cheating in a casino is generally illegal, however, it’s possible for a player to consistently win without cheating. The best-known example of this is card counting—a type of advantage play used in the Blackjack family of games which is not illegal and, in some cases, is even legally protected. This advantage play technique essentially takes the basic principles of the game of Blackjack and uses them against the casino, and these principles are fairly simple. So start a game, each player bets an amount of money, then, six decks of cards are shuffled together to form what’s called the shoe. Each player is dealt two cards, face-up, while the dealer gets one face-up and one face-down. The goal for all participants is simple—it’s to get their cards to total as close to 21 without going above 21. The execution of that is much tougher. Starting from the left, each player will either decide to stick with the total they have, or to take another card to add to it. Of course, the player doesn’t know what the next card will be worth, it could be anything from 1 to 11, so it’s a gamble on whether it’ll make the total go over 21—in which case their bet is lost. The higher the original total, the riskier it is to take another card, but there is, in fact, a mathematically optimal choice for every scenario. Once every player is done taking cards, or not, the dealer reveals the face-down card and, automatically, if their total is below 17, they take additional cards until it isn’t. If it’s 17 or higher, the leave it as is. There are then three scenarios. If the dealer goes above 21, all players’ bets are doubled, as long as they didn’t go above 21 first. If the player’s total is higher than the dealer’s, then the player’s bet is doubled. However, if the player’s total is lower, they lose their bet. Of course, this explanation skipped over plenty of smaller rules and unlikely edge-cases, but it is these fundamental elements of game-play that tie into why card counting works. Now, without getting too much into the math, on average, in Blackjack, higher-value cards benefit the player, while lower cards benefit the dealer. While the explanation for the higher-cards is more complex, lower cards benefit is based on the fact that they are required to take additional cards when their total is less than 17 and so a greater density of lower cards makes it less likely that they’ll total over 21—in which case each player’s bet is doubled. Therefore, if you know that a bunch of low cards are coming, you know that the odds are against you and so you should reduce your bet or not play. But, the question is, how do you know what’s coming in a randomly shuffled deck? Well, you perform process of elimination, or, even more simply, you count the cards you see. There are hundreds of different forms of card-counting that work in hundreds of different ways, but all are more or less based on what’s known as the Hi-Lo system. With this, each card is assigned a value. Two through six are assigned one, seven through nine are assigned zero, and ten, the face cards, and the ace are assigned negative one. This is based on the fact that, every time a high-value card is dealt, there are fewer of them in the deck, which means the odds get worse for the player considering that high-value cards are better for them, and vice versa. So, card counting is quite simple. With every card a player sees, they add up its assigned value. So, if there are three players, and they are dealt these cards, the running count would be one plus zero plus negative one plus one plus zero plus one plus zero, which would equal a total of two. That total of two indicates that the odds have shifted slightly in the player’s favor, while if it were negative two that would indicate the odds were in the dealer’s favor. As play goes on, and they get deeper into the deck, the running count will generally increase in one direction or the other, giving the player more confidence on where their odds stand and so if, for example, the running count equalled twenty, the player would know that the odds were greatly in their favor and therefore that they should bet big on the next round, as they have a greater than 50% chance of winning. This is how people can reliably make money in Blackjack. If a player changed their bet by a factor of fifteen depending on the odds, and the dealer waited until they’re through five of the six decks before shuffling, a player, following perfect Blackjack strategy, could earn an advantage of about 1.182% over the house. That means that if, assuming a table completes a round of play every minute and the average bet is around $200, a card counter could profit, on average, about $110 an hour—enough that some people can and do make a living by sitting at Blackjack tables, counting cards. However, considering how simple and reliable this advantage play method is, casinos go to great lengths to stop it, which is very, very difficult. That’s because the advantage is all in the mind—there’s no good way to fully prove someone’s card counting. Sometimes, people are just lucky, and it’s quite a bad look for casinos to kick people out just because they’re winning. That’s why, instead of trying to prove it, most casinos implement rules to try and stop card counting from working as well. Remember that, generally, the running count will get further into the positive or negative the further into the game one goes, because the card counter will have seen more cards that are now in the discard pile, and therefore cannot be dealt. While a few rounds in the running count might be in the single-digits positive or negative, further on, it’ll get into the double-digits which gives a counter great confidence on whether they should bet big or not. It’s towards the end of the shoe, the collection of un-dealt cards, when card counters really make their money, so to make it less profitable, casinos can just have their dealers shuffle earlier on. If they shuffle four decks deep into the six-deck shoe rather than five, that decreases the player’s advantage from 1.182% to just 0.568%. However, shuffling earlier and more often also cuts into the casino’s profits because, anytime the dealer is shuffling, the non-advantage players aren’t playing and losing money—which is how the casino makes its money. Another option for casinos is to increase the number of decks they shuffle together to make the shoe. Back before card-counting first became a widespread issue for casinos, they would play Blackjack with just a single deck of cards, but if they did this today, it would only be a matter of minutes before a card counter would have high confidence about the odds. Therefore, they typically now play with six decks shuffled together, which increases the time it takes to get to high confidence and, since time is money for a card counter, this decreases their profits. Some casinos take this a step further by using continuous shuffle machines. With no discard pile, there is no increase or decrease in beneficial cards in the shoe, so card counting is completely ineffective, however, these machines are not yet fully widespread due to distrust by frequent players. While these methods deal with stopping or reducing a player’s ability to actually know what the odds are at a given moment, the other method involves stopping a player’s ability to respond to this knowledge by changing the size of their bets. Essentially, if a dealer or a pit-boss suspects someone might be card counting, they’ll change the rules on them and require flat-betting. This is where a player is told to pick one bet size and then they are not allowed to change that size from round to round. Therefore, card counters might know that the odds are changing, but they will not be able to respond to it with a larger or smaller bet, so the odds will stay in the house’s favor. However, that poses the question, how do you spot a card counter? Well, one of the most tell-tale signs is that they’re wining. Even if someone is card counting, the casino does not care as long they’re losing, so they don’t really pay attention to people until they’ve made some real money. Once they are, though, if a pit-boss notices that a player, for example, changes their bet from $100 to $1,000 right before a series of wins or as the shoe is close to finished, that’s a good sign that they know what the odds are. In addition, professional card-counters often start with a very large buy in—they convert a lot of money into chips—because there is natural variance on whether or not they win, even if they know the odds. They can win infinite money, but they can only lose as much money as they have, and even if they have an advantage, if they’re unlucky and they lose all their money, there’s no way to win it back. The math works out so that, if you want to win $170 an hour card counting, you need to have $100,000 total in order to only have a 1% chance of running out of money. In the end, casinos don’t need to make it impossible to count cards. Only a small minority of people will attempt it and those that do can only make so much money per hour, so in a way, its a cost of doing business. To avoid it making a big dent in their profits, all they need to do is make advantage play at their tables just a little bit harder than those next door and, if this is the case, the card counter will go next door. There will always be an escalating arms race by both players and casinos to gain an advantage and stop an advantage, respectively. Mathematics, economics, and human nature combined mean that as long as Blackjack and other flawed casino games stay popular, players will always find a way to tilt odds ever so slightly in their favor. 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