Shamal Lalvani, Aggelos Katsaggelos
©SHUTTERSTOCK.COM/NETFALLS REMY MUSSER
In movies, blackjack card counters confidently place huge bets in casinos when the time is just right, making huge profits. While counting cards can be used to profit from casinos, the truth of how card counters profit is a bit different. In this article, we get to the heart of how casinos and card counters manage their profits—the law of large numbers, a fundamental theorem in statistics. We discuss how card counting works and what this looks like in practice.
In 1966, mathematics professor Edward Thorp published the book Beat the Dealer. In this book, he mathematically analyzed the game of blackjack and provided a winning strategy with which a blackjack player could profit from casinos. In fact, he profited from his strategy, too. He had many adventures at casinos, including building a wearable computer with Claude Shannon (the father of information theory), which they used to perform calculations at roulette wheels in casinos. After Thorp fattened his wallet at the casinos, he moved on to Wall Street—what he considered the “greatest casino on Earth.” Currently, Thorp is a hedge fund manager and has a personal net worth of about US$800 million. He recounts that he learned valuable lessons in investing from his experiences at the casinos.
So, how does one maximize their profits in blackjack? It depends on the rules that are offered at the casino. Blackjack offered at casinos is a different game today than it was at the time of Edward Thorp. For example, if the deck is shuffled after each hand, then the optimal strategy is not to play at all. However, if the deck isn’t shuffled after each hand, and other specific criteria are met for the game (which typically vary with casinos, such as the payout for getting a blackjack), the optimal strategy is to “count cards.” Counting cards uses the simple observation that future hands in blackjack are not independent of previous hands because the player observes the previous cards that have been dealt (unless the shoe—a collection of decks used for dealing blackjack—is freshly shuffled). We discuss more of this later, but, first, let’s review how blackjack works.
Blackjack is a game offered at most casinos where the goal is for the player to beat the dealer. A player places a bet before each round of blackjack. Two cards are dealt to the player face up, and two cards are dealt to the dealer, one face up and one face down. Cards are assigned their numerical value, where face cards (the jack, queen, and king) are assigned a value of 10, and aces can be counted as either a one or an 11 (whatever the player prefers). The goal is to get as close to 21 as possible without going over 21.
After the cards are dealt, the player may ask for additional cards or stay at their current value. If the player goes above 21, they “bust” and lose their bet. Then, the dealer reveals their hidden card and keeps dealing until a minimum of 17 is reached. If the player has a higher value than the dealer, the player wins and is returned double their bet. If the player and the dealer have the same value, then the player keeps their original bet. This is known as a push. The exception to a “push” is blackjack, where the player has an ace and a face card and instantaneously wins before the dealer’s cards are revealed.
However, rules vary at different casinos, and so do payouts for blackjack. There are additional rules a player can invoke in blackjack, such as “splitting,” where they play two hands, one for each card they are dealt. Splitting can only be invoked when the dealer is dealt two cards of the same type (for example, two kings or two aces). Players may also “double down,” where they receive only one more card but may double their bet. Before we talk about counting cards, let’s talk about the law of large numbers.
Informally, the law of large numbers simply states that, if we collect a very large sample from a certain probability distribution (where samples are independent of each other), the mean of our sample should come close to the mean of the probability distribution. For example, consider flipping an unbiased coin, where the probabilities of heads and tails are both one half. If we repeat the process of flipping such a coin many times, we would expect about half of our flips to land on heads and the other half of our flips to land on tails. This is the law of large numbers in action. Figure 1 shows the number of flips and the percentage of times the coin lands on tails from a simulation. Notice that, as our sample size gets larger, we approach the true mean of the probability distribution—mainly, about 50% of tails flipped.
Fig 1 The law of large numbers for flipping an unbiased coin.
The law of large numbers may not seem special at first glance, but it is fundamental for a lot of the tools we use in statistics and stochastic processes. Casinos manage their operations through the law of large numbers. For example, slot machines are designed both to be volatile and to produce randomized outcomes (with standards set by gaming commissions to protect players from “unfair” slot machines). However, slot machines are also designed in such a way that the mean return per spin (for the casino) is positive (usually 5%–10%). On any single day, a given slot machine may pay players more money than it takes in, resulting in a loss for the casino. However, in the long run, over millions and millions of spins of the machine, the law of large numbers guarantees that the casino gets quite close to the true mean return—about 5%–10% of the total money that was bet on the machine. Now that we know how casinos profit from the law of large numbers, we see that players, if they consistently play for a long period of time, lose their money for the same reason. Playing a game with a negative expected value (–5% to –10% for slot machines) has no winning strategy in the long term.
Blackjack rules are specifically set in casinos to yield a payoff of between 1% and 2% per hand for the casino, specifically when the players play the optimal strategy. If players do not play the optimal strategy, the payoff can be larger for the casino. The rules are typically set through Monte Carlo simulations, where the payoffs over many hands of blackjack can be studied (again, here comes the law of large numbers in action), so the casino can ensure they will profit in the long run.
Since we see that blackjack is a game with a negative expected value overall for the player, it might seem that players should not play at all. However, as mentioned before, hands are not independent of each other, and we observe previous cards dealt when playing blackjack. The game of blackjack can be a positive-expected-value game for the player based on the previous cards dealt. The idea is to bet larger when the expected value of the game is positive and lower (or not at all, if possible; this strategy is known as wonging) when the expected value is negative.
Counting cards allows players to gauge if the game has a positive or negative expected value based on the previous cards dealt. The most common card-counting strategy is the “hi–lo strategy.” The player assigns a value of –1 to the high cards that have been dealt (aces, tens, jacks, queens, and kings) and a value of +1 to the low cards that have been dealt (two through six). All other cards are assigned a value of zero. The player keeps a running tally of the “running count” of all of the cards that have been dealt. For example, if two low cards and three high cards have been dealt, the count is 2 − 3 = −1. Therefore, when the player has a high count, it means that the remaining cards to be dealt are rich in high cards—cards that are more likely to give the player blackjack. That is when the player bets bigger to maximize their profits. When the “true count” is below a specific threshold (the true count takes into account the number of decks and remaining cards in play in the shoe to normalize the count with respect to one deck), the optimal strategy is not to play at all.
We have seen that the game of blackjack can be a game with a positive and negative expected value based on the “count.” The strategy for the player is to bet high when the count is high and bet low when the count is low. In practice, it is worth noting that, even when the count is high, the player typically sees about a 51% chance of winning their next hand. Therefore, it does not make sense to bet all of one’s money on the next blackjack hand. The idea is that, over many hands of blackjack, they should expect to win about 51% of blackjack hands when the count reaches a specific value. If one were to bet all their money when the count is high, they might lose it all and no longer be able to make profits, even when the count is high.
How do card counters determine how much to bet? They must take the variability of the game into account. A typical rule is to bet around 1% of their “bankroll,” but betting software exists that allows card counters to determine how much to bet. It is worth noting one more theorem that comes into play—the central limit theorem. The central limit theorem, informally, tells us that, if we collect a large sample of independent draws from a specific probability distribution, the mean of our sample is normally distributed. The magic here is that it does not matter what probability distribution we are sampling from. It is worth noting that the central limit theorem can be derived from the law of large numbers. The central limit theorem can be used to calculate risk in card counting based on how many hands are played. Figure 2 shows a simulation of 5,000 sample means drawn from 1,000 flips of a coin. Notice the bell-shaped curve, characteristic of a normal distribution.
Fig 2 The central limit theorem for samples of an unbiased coin.
One last thing to note is that card counters typically must understand the rules of the game of blackjack that is being offered to know if it is even worth it to play. For example, some games of blackjack use automatic shuffling machines, which makes counting cards impractical. Other games have low deck penetration, where a low number of cards are dealt from the shoe before it is reshuffled. For example, a shoe with six decks and 50% deck penetration usually offers a break-even game.
In the United States, counting cards is not illegal (unless one uses a device to assist them in counting the cards). Nevertheless, casinos will typically ask card counters to stop playing the game or leave the establishment once they figure out a player is counting. Although it is a legal gray area if casinos have the right to ask card counters to leave their establishment simply for counting, businesses usually reserve the right to refuse service to any person for any reason and may deny future entry. Nevertheless, many people have made livings by counting cards. One popular team was the “Church Team,” which organized a card-counting group that profited significantly.
Card counters usually play a “cat-and-mouse game” with casinos, where they try to avoid detection from casinos, and casinos respond by making it harder for players to count cards. To illustrate, changes have been made to blackjack offered in casinos since the time of Edward Thorp. Casinos used to offer single-deck blackjack, but now two to eight decks per shoe is standard. Additionally, the rules have been modified over time to bring a higher house edge to casinos. Casinos commonly monitor players through video cameras and ask players whom they suspect of counting cards to leave. They also share the names of card counters with other casinos so that other casinos may stop them from playing. Card counters commonly try to avoid detection by keeping a smaller bet spread and not giving casinos their IDs when possible so that other casinos will not have their names to prevent them from counting cards.
In summary, the law of large numbers is a simple but powerful tool for maximizing profits for betting strategies, over the long term, where one’s return typically converges to the mean return. In fact, the law of large numbers is not only used in blackjack but in all forms of “advantage play”—games offered in casinos where players find a positive expected value.
• C. Jones, The 21st-Century Card Counter: The Pros’ Approach to Beating Today’s Blackjack. Las Vegas, NV, USA: Huntington Press, 2019.
• E. O. Thorp, Beat the Dealer. Las Vegas, NV, USA: Huntington Press, 1966.
• B. Carlson, Blackjack for Blood. New York, NY, USA: Vintage, 2017.
• S. M. Ross, A First Course in Probability. Upper Saddle River, NJ, USA: Prentice-Hall, 2006.
Shamal Lalvani (shamal.lalvani@northwestern.edu) earned his M.S. degrees in mathematics and operations management from Miami University and the Kellogg School of Management at Northwestern University, respectively. He is a Ph.D. student in electrical engineering at Northwestern University, Evanston, IL 60208 USA. His research interests include combining machine learning with neuroscience and psychological models of decision making. Prior to his Ph.D., he was a blockchain developer at Tata Consultancy Services.
Aggelos Katsaggelos (a-katsaggelos@northwestern.edu) earned his M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology prior to joining the Department of Electrical Engineering and Computer Science at Northwestern University. He has authored considerably in the areas of multimedia signal processing and communications, computational imaging, and machine learning. Among his various professional activities, he was the editor-in-chief of the IEEE Signal Processing Magazine (1997–2002). He is a Life Fellow of IEEE.
Digital Object Identifier 10.1109/MPOT.2023.3276572