Blackjack Probability and Blackjack Odds

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To fully understand the game of blackjack, you must understand and master blackjack odds. It's crucial to know how the casino gains its edge and how it helps them win. It's also important to understand blackjack odds such as the odds of hitting a 10 or the odds of being dealt a blackjack. You can read over these blackjack odds charts to help understand the logistics behind blackjack.

Casino and Dealer Advantage in Blackjack

In most casino games of blackjack, the house advantage (the dealer advantage) is ~ 8%. The house gets this advantage by the dealer being the last player to act. By acting last, all other players have already made their decisions and could quite possibly bust before the dealer has his turn.

By using correct blackjack basic strategy, you can turn the casino edge in blackjack from 7%-8% down to 0.5%. If you correctly use advanced card counting techniques, you can often change blackjack odds and give yourself the advantage over the casino. Manipulating the house odds to your favor is the reason most casinos don't allow card counters to play blackjack.

Probability of Busting on a Hit

It's very important to know the probability of your hand busting when you are holding any total in the game of blackjack. The following odds chart shows the blackjack odds of busting, depending on your current hand value:

Two-Card Count Frequencies

This interesting blackjack odds chart is the two card count frequency chart. This chart shows the percentage chance that you will be dealt a hand in each given value range. The most important frequencey to note is the chance of being dealt a natural blackjack (natural 21 value. The odds of being dealt a natural blackjack are merely 4.8%. Following this chart you will see that the most common two card hand, at 38.7%, is a hand totaling 1-16, which is considered a decision hand.

Dealer Final Hand Probabilities

Blackjack Standard Deviation

This blackjack odds chart shows the dealer final hand probability. These are the percentages that the dealer will end up with a hand totaling each corresponding value (up to 16). Read over this chart to understand the odds that the dealer has to make his final hand.

Player Advantage vs. Dealer Up Card

The first two columns in this odds chart explain the dealer's chance of busting, depending on the up card that he is showing. You should note that the dealer has the highest chance of busting when he is showing a 5. The third column in this chart shows the player advantage of using basic strategy, compared to each up card the dealer is showing. You can see that the player has the highest advantage of 23.9%, when the dealer is showing a 5. When the dealer is showing any card that is 9 value or higher, the player is in the negative advantage range.

Effects of Removing Cards from a Deck

When looking at the odds of removing certain cards from a 52-card deck, some cards have a much greater effect on blackjack odds. To create the strongest card counting system ever invented, you would have to incorporate all of these slight and subtle differences into the numbers to be a completely accurate system.

Removing every 5 from a deck cards would make the largest impact of improving your blackjack odds, as a player. On the other hand, removing every Ace from a deck of cards would make the largest impact on improving the odds for the casino.

See also:

• Appendices
• Miscellaneous
• External Links

Introduction

This appendix presents information pertinent to the standard deviation in blackjack. It assumes the player is following basic strategy in a cut card game. Each table is the product of a separate simulation of about ten billion hands played.

As a reminder, if the variance of one hand is v, the covariance is c, and the number of hands played at once is n, then the total variance is n×v + n×(n-1)×c.

The following table is the product of many simulations and a lot of programming work. It shows the variance and covariance for various sets of rules.

Summary Table

By way of comparison, Stanford Wong, in his book Professional Blackjack (page 203) says the variance is 1.28 and the covariance 0.47 for his Benchmark Rules, which are six decks, dealer stands on soft 17, no double after split, no re-splitting aces, no surrender. The second row of my table shows that for the same rules I get 1.295 and 0.478 respectively, which is close enough for me.

Effect on Variance of Rule Changes

The next table shows the effect on the expected value, variance and covariance of various rule changes compared to the Wong Benchmark Rules.

Effect of Rule Variation

What follows are tables showing the probability of the net win for one to three hands under the Liberal Strip Rules, defined above.

Liberal Strip Rules — Playing One Hand at a Time

The first table shows the probability of each net outcome playing a single hand under what I call 'liberal strip rules,' which are as follows:

• Six decks
• Dealer stands on soft 17 (S17)
• Double on any first two cards (DA2)
• Double after split allowed (DAS)
• Late surrender allowed (LS)
• Re-split aces allowed (RSA)
• Player may re-split up to three times (P3X)

6 Decks S17 DA2 DAS LS RSA P3X — One Hand

The table above reflects the following:

• House edge = 0.28%
• Variance = 1.303
• Standard deviation = 1.142

Probability of Net Win

I'm frequently asked about the probability of a net win in blackjack. The following table answers that question.

The next three tables break down the possible events by whether the first action was to hit, stand, or surrender; double; or split.

Blackjack Standard Deviation Calculator

Net Win when Hitting, Standing, or Surrendering First Action

Net Win when Doubling First Action

Net Win when Splitting First Action

Blackjack Standard Theory

Liberal Strip Rules — Playing Two Hands at a Time

The following table shows the net result playing two hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the two hands.

6 Decks S17 DA2 DAS LS RSA P3X — Two Hands

The table above reflects the following:

• House edge = 0.28%
• Variance per round = 3.565
• Variance per hand = 1.782
• Standard deviation per hand= 1.335

Liberal Strip Rules — Playing Three Hands at a Time

The following table shows the net result playing three hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the three hands.

6 Decks S17 DA2 DAS LS RSA P3X — Three Hands

The table above reflects the following:

• House edge = 0.285%
• Variance per round = 6.785
• Variance per hand = 2.262
• Standard deviation per hand= 1.504

Internal Links

Blackjack Standard Strategy