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Probability in Blackjack
Discrete Mathematics · Chapter 7: Probability in Games, Gambling, and Strategic DecisionSection 7.2 · lesson 41 of 8710 min · free with a free account
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In this lesson
A finite deck, cards removed one at a time, and exact numbers instead of a hunch.
- Model a shuffled deck as a uniform sample space
- Count a natural blackjack correctly
- Update probabilities as cards are removed
- See why bust probability alone doesn't decide hit or stand
Transcript
Everything said in the video, section by section (1,457 words). Read along, or search it.
Probability in Games, Gambling, and Strategic Decision
Probability in Blackjack
This section applies discrete probability to blackjack. We'll treat the deck as a finite sample space, count hands correctly, and update probabilities as cards are removed. And we'll see why the chance of busting, on its own, doesn't settle whether to hit or stand.
The Opening Problem
Hard 16 Against a Dealer 7
A player at a blackjack table is looking at a hard sixteen, a ten and a six, face up. The dealer shows a seven. The player has a few seconds to decide. Hit, or stand? That decision rests on the same discrete probability machinery as a basic-strategy card, an advertised house edge, and the random number generator behind an online table. Treat the deck as a finite, countable sample space. Update probabilities as cards are removed from it. And compare exact numbers, rather than a hunch.
7.2.1 The Deck
52 Cards, 13 Ranks, 1 Flexible Ace
A standard deck has fifty-two cards, in thirteen ranks across four suits. Suits don't affect play. Every card is worth its number, except the face cards, jack, queen, and king, worth ten each. Together with the tens, that's sixteen ten-value cards. And the ace is worth one or eleven, whichever helps more without going over twenty-one. A hand using an ace as eleven is called soft. Ace and six is a soft seventeen. Once the ace can only count as one without busting, the hand is hard.
Worked Example 7.2.1
Counting Every 2-Card Hand
Dealing a hand selects cards from a finite population, without replacement. If the deck is well shuffled, every card is equally likely in every position. So the deck is a discrete uniform probability space. The probability of an event is the number of favorable outcomes, over the total. How many two-card starting hands are there? Order doesn't matter. A king then a six is the same hand as a six then a king. So it's a combination. Fifty-two choose two is fifty-two times fifty-one, over two. That's one thousand three hundred twenty-six hands. That's the denominator for every probability from a fresh two-card deal.
Worked Example 7.2.2
The Chance of a Natural
A natural, or blackjack, is a first two-card hand totaling twenty-one. One ace, and one ten-value card. There are four aces, and sixteen ten-value cards. A natural takes one card from each of those two separate groups. So by the multiplication rule, there are four times sixteen, sixty-four naturals. Here's a common error. It's not twenty choose two. That would count every pair from the combined pool of twenty, including ace-ace and ten-ten pairs, and neither is a blackjack. So the probability is sixty-four over one thousand three hundred twenty-six, which is thirty-two over six hundred sixty-three. About four point eight three percent. That's roughly one natural every twenty point seven hands, on average.
Definition 7.2.2
Drawing Without Replacement
That was an unconditional probability, describing a hand before any cards are known. Once the player's cards and the dealer's up card are revealed, those cards leave the unseen population. Every later probability has to be recomputed on what remains. Drawing without replacement from a finite population is modeled by the hypergeometric distribution. A population of N items has K successes. If n items are drawn without replacement, the probability of exactly k successes is K choose k, times N minus K choose n minus k, over N choose n.
1 Card at a Time
Bad Cards Over Cards Left
Now, will the single next card bust my hand? That's exactly one success, in a draw of exactly one card, from N unseen cards, where K counts the busting cards. Substitute n equals one and k equals one. K choose one is K. N minus K choose zero is one. And N choose one is N. So the probability is simply K over N. The familiar rule, count the bad cards and divide by the cards remaining, isn't an intuitive shortcut. It's the hypergeometric formula, evaluated at one draw, and every step is exact.
Worked Example 7.2.3
P(Bust) on Hard 16 Against a 7
Back to the hard sixteen against a dealer seven. If the player hits, what's the exact chance of busting, from a single freshly shuffled deck? First, remove the known cards. The player's ten and six, and the dealer's seven, leave forty-nine unseen cards. Second, find the busting ranks. From sixteen, a card worth six or more busts. That's six, seven, eight, nine, and the ten-value group. A five makes exactly twenty-one. And an ace automatically counts as one when eleven would bust, so it makes seventeen. Third, count what's left of those ranks. Three sixes and three sevens, since one of each is already showing. Four eights, four nines, and fifteen ten-value cards. That's twenty-nine. So the probability of busting is twenty-nine over forty-nine, about fifty-nine point two percent.
Figure 7.2.1
The 49 Unseen Cards
This chart shows all forty-nine unseen cards, grouped by rank. The gold bars are the twenty-nine that bust a hard sixteen. The green bars are the twenty that don't. It shows why two things can be true at once. Most of the cards are high, so busting is more likely than not. But it's fifty-nine percent, not a certainty.
Hit or Stand?
Bust Risk Isn't the Whole Comparison
Fifty-nine percent looks like a strong argument for standing. But basic strategy says hit, in exactly this spot. The relevant comparison isn't the bust probability in isolation. It's the probability of losing the hand, under each choice. A dealer showing a seven must keep hitting until reaching seventeen. And a standing sixteen loses to any dealer total that doesn't bust. Hitting, despite its own high bust risk, produces fewer losses overall than standing does. A basic-strategy chart is exactly that comparison, made for every hand total against every dealer up card.
Code Example 7.2.1
Bust Probability as a Function
The bust calculation turns directly into reusable code. The function removes the known cards from the counts. Then it adds up the remaining cards that would push a hard total past twenty-one, counting an ace as one, and divides by the unseen total. Running it on a hard sixteen, with a ten, a six, and a seven removed, prints zero point five nine one eight. It confirms the hand calculation.
7.2.4 Where Basic Strategy Comes From
A Branching Tree, Worked Out
That function answers one question, for one hand. A full chart also needs the dealer's whole outcome distribution, for every up card. The dealer's play is a fixed rule. Hit on sixteen or less, and stand on seventeen, though some tables hit a soft seventeen. Only the player makes a choice. But each dealer hit branches into every possible next card, and each branch can branch again. The number of sequences multiplies with every hit. So modern software evaluates that tree with recursive methods, or approximates it by simulating many shoes. This isn't a new problem. In nineteen fifty-six, four U.S. Army mathematicians at Aberdeen Proving Ground published the first mathematically derived basic strategy, worked out by hand. And in nineteen sixty-two, Edward Thorp's book, Beat the Dealer, refined it with an electronic computer.
Card Removal, Both Ways
Counting Cards - and Countering It
Thorp also showed something more. Tracking how the deck's remaining composition shifts as cards are removed, across a whole shoe, can tilt the game's long-run expectation toward the player. That's exactly the without-replacement reasoning of this section. And it's what card counters exploit. Casinos target that dependency directly. Many tables use continuous shuffling machines, which return each round's cards to the shoe and reshuffle immediately. That keeps the unseen population close to full on every hand, pushing the game back toward draws with replacement, and undercutting what a counter relies on. And regulated online tables use certified random number generators. The same uniform model, enforced by software.
Table 7.2.2
Basic Strategy, Hard 13 to 16
This table shows basic strategy for hard totals thirteen through sixteen, with each hand's baseline bust probability from full-deck proportions. The bust chance climbs from thirty-eight and a half percent at thirteen, to sixty-one and a half at sixteen. Yet the rule is the same for all four. Stand against a dealer two through six, and hit against a seven through ace. Hard twelve is left out on purpose. It's the well-known exception, hitting against a two or three, but standing against four through six.
Takeaways
Exact Numbers, Shrinking Deck
To summarize. A well-shuffled deck is a discrete uniform sample space. Two-card hands are counted with fifty-two choose two, one thousand three hundred twenty-six. Counting correctly matters. A natural is four times sixteen, sixty-four hands, not twenty choose two. Once cards are known, probabilities are conditioned on the smaller population. That's the hypergeometric distribution, and for one card it's K over N. And the hit or stand decision compares the chance of losing after hitting, with the chance of losing after standing. Here's something to think about. A hard sixteen busts far more often than a hard twelve. Yet strategy hits sixteen against a seven, and stands on twelve against a four, five, or six. Explain why bust probability alone isn't enough to decide. What second probability has to be weighed against it? Here's a hint. What happens to a hand that stands, but gets beaten anyway? And how often does that happen against a weak dealer card, compared with a strong one?
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