Beat the Dealer: A Winning Strategy for the Game of Twenty-One
Edward O. Thorp, 1962
It seems that luck rules in a casino: one deal has nothing to do with the next. But in blackjack the cards come from a finite deck and don't go back, so the odds shift from hand to hand. Edward Thorp shows how to play by a chart, count the remaining cards and check your calculation, and why even a precise system is powerless against a crooked game and a short run of bad luck.
Who it is for
For anyone curious how probability changes decisions at the card table, and why an edge on paper doesn't rule out the risk of losing.
“In the modern casino game of blackjack, the player can gain a consistent advantage over the house by using the strategy presented in this book.”
Ten silver dollars and one strange move
Over Christmas vacation, the mathematician Edward Thorp went with his wife to Las Vegas, Nevada. Gambling held no interest for him.
Before the trip, Professor Sorgenfrey had told him about an article in a mathematics journal that described a blackjack strategy and claimed it left the casino an edge of 0.62 percent. A note in the book explains that, because of an arithmetic error, the article gave 0.62 percent instead of 0.32, and that the correct figure for this strategy is a player's edge of 0.09 percent. Thorp copied the strategy onto a little card.
At first Thorp's strange moves drew condescending pity from the bystanders, but a seven-card 21 surprised them. Someone said Thorp had a $25 bonus coming, but the dealer explained that such a bonus was paid only at a few casinos in Reno. Fifteen minutes later Thorp stopped, having lost eight and a half of his ten silver dollars.
This failed test left a question: if even experienced players don't know the fundamentals of the game, could there be a better way to play than by instinct? Thorp went home and started calculating. The book shows how he built his strategy and where its power runs out.
The right move sometimes looks like madness
One of the most awkward-looking hands came later, once Thorp had learned to count cards. He held a soft 17: an ace and a six, which for now made a total of 17. The dealer showed a seven, and the remaining deck gave Thorp an edge of about five percent. Thorp put down the table maximum of five hundred dollars.
To the onlookers the hand seemed all but finished, and drawing looked dangerous. But his chart told him to go on.
First Thorp got an eight, which turned his soft 17 into a hard 15. Then came an ace, which kept him from going bust and gave him 16. He drew one more card, a three, and stood on 19. The dealer turned up an ace as his hole card beneath the seven and, by the rules, had to stand on 18. A decision that looked like a risk of ruining a decent hand turned out to be the only way to win that deal.
Basic strategy doesn't try to guess the next card. It weighs the total of your cards against the dealer's up card, relying on probabilities worked out in advance. If you have a soft hand, the ace can count as one, so an extra card isn't always as dangerous as it seems. A hard hand gives you no such cushion. That's why the same totals sometimes call for different moves.
Before Thorp started counting the cards dealt, he mastered the basic decisions for every hand: when to draw, when to stand, when to double down and when to split a pair. He didn't play by the simple rule of “don't go over.” A player who copies the dealer and never doubles down or splits pairs loses to the casino, by the book's calculations, an average of 5.6 percent of the money bet.
The chart is like a map of forks in the road: it doesn't know which card will come next, but it compares the possible outcomes of each move. Its anchor is the dealer's up card, the only card of your opponent's you can see before deciding. Doubling down and splitting pairs are decisions you make by calculation, not out of fear.
How to apply: first work out the total of your hand and look at the dealer's up card. Then check the move against the chart for a hard or soft hand. The calculation changes under other table rules, for example if the dealer draws on a soft 17 or doubling down is not allowed. Under the typical rules Thorp gives, basic strategy produces a result of about zero: a small edge may lie with either the casino or the player. It cuts down on mistakes, but on its own it doesn't promise a win.
About the author
Edward O. Thorp is an American mathematician, professor and investment fund manager who applied probability theory to blackjack and popularized card counting.
All books by this author: Edward O. Thorp
Terms from the book
- Blackjack
- Braun
- Expected value
- Salmon
- Reno: A city in Nevada where Thorp tested his blackjack system.
- Basic strategy: A chart of decisions for every player's hand against every dealer's up card. It is worked out for a given set of rules and ignores the cards dealt in earlier rounds.
- Dealer's up card: The dealer's card that everyone can see before the player decides whether to draw, stand, double down or split a pair.
- Soft hand: A hand with an ace that can count as 11 without going over 21. If the next card is dangerous, the ace can be counted as 1 instead.
- Hard hand: A hand without an ace that can count as 11. Its total can't drop by ten points after a new card.
- Bust: A card total above 21. A player who busts loses the bet at once, even if the dealer later goes over 21 too.
- Doubling down: The decision to double the original bet and receive just one more card. Under the book's usual rules it can be made after the first two cards are dealt.
- Splitting pairs: The decision to turn two matching starting cards into two separate hands and add a second bet to them.
- Independent trials: Repeated events in which the result of each one doesn't change the probability of the next. Ordinary rolls of a fair die and spins of a roulette wheel work this way.
- Five-count: A simple way to track whether all four fives have left the deck. In Thorp's system it is a rare signal of a favorable deal.
- Point count: A system that gives small cards positive values and large cards negative ones, and uses the total to judge the rest of the deck.
- Running count: The sum of the values of the cards seen so far. It changes as the cards are dealt and resets once the used cards are shuffled back in with the rest.
- High-low index: A normalized estimate of the deck's makeup in Thorp's full system: the point total is divided by the number of cards not yet seen and the result is multiplied by 100. The index is expressed as a percentage.
- Others/tens ratio: The ratio of the remaining cards that aren't tens to the cards worth ten. In Thorp's system a full starting deck gives a ratio of 36 to 16, or 2.25.
- Insurance: An extra bet the casino offers when its up card is an ace. It wins if the dealer's face-down card is worth ten.
- Nevada: The US state that is home to Las Vegas and Reno. Thorp ran an early test of his system there with money from two investors.
- Bankroll: The sum set aside for playing. Thorp warns: never play with money whose loss would hurt you.
- Puerto Rico: An island in the Caribbean where, at the time Thorp describes, casinos ran under government supervision and often dealt two decks down to the end.
- End play: The last deals before the cards are shuffled. Under some old rules a player could change the number of hands he played and influence which cards stayed out of the next deck.
All terms
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