# Probability and Expected value in Blackjack

In the world of online casinos, blackjack stands out as a game where skilled players can significantly reduce the house edge through optimal play. This is true whether you're playing at a traditional casino or enjoying online blackjack from the comfort of your home. To achieve success in both online and offline blackjack, understanding two key mathematical concepts is crucial: probability and Expected Value (EV).

This article will explore both concepts, demonstrating how they form the foundation of expert blackjack strategy. Whether you're a casual online blackjack player or aspiring to master the game, grasping these mathematical principles can improve your play and help you make better decisions with various Blackjack hands.

We also have a dedicated Blackjack guide in which we explain different concepts and strategies you should take into account when playing any Blackjack game.

## Basic Probabilities in Blackjack

The probabilities in blackjack depend on the number of decks used or if you play a multi-hand game. Let's examine some key probabilities for games using one or two decks:

### Probability of Initial Hands (First Two Cards)

The first two cards dealt in blackjack set the stage for the entire hand. We'll break down the probabilities of various starting hands, from the coveted natural blackjack to common two-card totals, providing you with valuable insights to inform your gameplay.

#### Natural Blackjack

- 1-deck game: 1.20663%
- 2-deck game: 1.19492%

#### Any Blackjack (including natural)

- 1-deck game: 4.82654%
- 2-deck game: 4.77968%

#### 20 points

- 1-deck game: 10.25641%
- 2-deck game: 10.45556%

#### 19 points

- 1-deck game: 6.03318%
- 2-deck game: 5.97460%

#### 18 points

- 1-deck game: 6.48567%
- 2-deck game: 6.4973%

#### 17 points

- 1-deck game: 7.23981%
- 2-deck game: 7.16952%

#### Probability of a "good" initial hand (Blackjack, 20, 19, or 18)

- 1-deck game: 27.60180%
- 2-deck game: 27.70724%

## Calculating Probabilities During the Game

For games played with complete decks, the probability of drawing a specific card value can be calculated using the following formula:

- P(x) = (4m - n(x)) / (52m - N) for cards with values 2-9 and Ace
- P(x) = (16m - n(x)) / (52m - N) for cards with value 10 (including face cards)

Where:

- m is the number of decks
- n(x) is the number of cards with value x already dealt
- N is the total number of cards already dealt

## Understanding Expected Value (EV)

Expected Value, also known as expectation, long-run expectation (LRE), fair value, or simply EV, is a fundamental concept in blackjack strategy. It represents the average outcome if a specific decision were made many times under identical circumstances.

### EV in Action: A Practical Example

Let's consider a real-world scenario:

You bet $1,000 and are dealt two picture cards (a total of 20), while the dealer shows a 6 upcard. What's the fair value of this hand?

Possible outcomes:

- Dealer draws to 21, you lose (9.7% chance)
- Dealer draws to 20, push (10.2% chance)
- Dealer gets 19 or worse, you win (80.1% chance)

Calculating the EV:

- 80.1% of the time, you win $1,000
- 10.2% of the time, you neither win nor lose
- 9.7% of the time, you lose $1,000

The net result: On average, you win $704 for each time you get this situation. The EV of this hand is +70.4% or $704.

## Using Probability and EV to Make Decisions

Understanding both probability and EV allows players to make mathematically sound decisions. Here are two common scenarios:

### Splitting Tens vs. Standing on 20

Some players like to split tens when the dealer has a six. Let's compare:

- Standing on 20 against a dealer's 6: EV = +70.4%
- Splitting tens: Each ten against a dealer's 6 has an EV of +28.8%

Two split tens (2 x 28.8% = 57.6%) is less valuable than one hand of 20 (70.4%), so it's better to stand on 20 than split the tens.

### Hitting or Standing on 12 vs. Dealer's 3

• Hitting 12 vs. dealer's 3: EV = -23.3%

• Standing on 12 vs. dealer's 3: EV = -25.2%

While both options have negative EV, hitting is slightly better as it loses less on average.

## Dealer's Probability Distribution

Understanding the dealer's probability distribution is key to optimal play:

- • Result: 17 – Probability 0.145
- • Result: 18 – Probability 0.139
- • Result: 19 – Probability 0.133
- • Result: 20 – Probability 0.180
- • Result: 21 – Probability 0.072
- • Result: Blackjack – Probability 0.047
- • Result: Bust – Probability 0.281

The dealer will bust about 28.1% of the time, or more than one in four hands on average.

## Optimal Fixed Strategy

Based on these probabilities and EV calculations, an optimal fixed strategy for a player, without card counting, involves:

- • Against dealer's 4, 5, or 6: Draw only to 11
- • Against dealer's 2 or 3: Go to 13 and above
- • Against dealer's 7 to Ace: Draw to 17

This strategy can reduce the house edge to between 0.64% and 0.88% of the initial wager, depending on the specific rules of the game.

## Advanced Strategy: Card Counting

Card counting systems, like the High-Low count, can further improve a player's odds by adjusting strategy based on the composition of the remaining deck. These systems essentially attempt to track changes in probabilities and EV as cards are dealt.

## Conclusion

The mathematics of blackjack, particularly probability and Expected Value, form the foundation of optimal play. By understanding these concepts, players can make informed decisions that minimize the house edge and maximize their chances of success.