What Is Expected Value (EV)?

Expected Value (EV) is a concept from statistics used to measure the average amount one can expect to win or lose if a particular event is repeated many times. In games like Baccarat, EV is used to evaluate the average return of different betting options, helping players make more informed and favorable betting decisions.

Mathematical Definition of Expected Value:：

The formula for expected value is as follows:

EV = (Potential Gain per Bet × Probability) − (Potential Loss per Bet × Probability)

• EV: Stands for Expected Value.
  
• Potential Gain: The amount of winnings the player receives. Potential Loss: The amount the player bets (typically equal to one betting unit).
  
• Probability: The likelihood of the outcome occurring.

In Baccarat, the expected value for different betting options can be calculated using the formula:

EV = ∑(Odds × Win Probability) − (Bet Amount × Loss Probability)

Application of Expected Value in Baccarat

The following are EV calculations for the main betting options in Baccarat, based on a standard 8-deck shoe:

1. Banker Bet
   
   • Odds: 1:0.95
   • Win Probability (Banker): 45.86% (0.458597)
   • Lose Probability: 54.14% (1 − 0.458597)
   • Calculation:
   EV = (0.95 × 0.458597) − (1 × 0.541403)
   EV = 0.435667 − 0.541403 = −0.010579
   Result: For every 1 unit bet, the average loss is about 0.010579 units. This reflects a house edge of approximately 1.06%.
   
   
2. Player Bet
   
   • Odds: 1:1
   • Win Probability (Player): 44.62% (0.446247)
   • Lose Probability: 55.38% (1 − 0.446247)
   • Calculation:
   EV = (1 × 0.446247) − (1 × 0.553753)
   EV = 0.446247 − 0.553753 = −0.107506
   Result: For every 1 unit bet, the average loss is about 0.107506 units. This reflects a house edge of approximately 1.24%.
   
   
3. Tie Bet
   
   • Odds: 1:8
   • Win Probability (Tie): 9.52% (0.095156)
   • Lose Probability: 90.48% (1 − 0.095156)
   • Calculation:
   EV = (8 × 0.095156) − (1 × 0.904844)
   EV = 0.761248 − 0.904844 = −0.143596
   Result: For every 1 unit bet, the average loss is about 0.143596 units. This reflects a house edge of approximately 14.36%.

Key Characteristics of Expected Value

1. Positive vs. Negative EV:
   • Positive EV: Indicates a profitable bet over the long term.
   • Negative EV: Indicates an expected long-term loss — the foundation of how most casino games are designed.
   
   
2. Long-Term Effects：
   The outcome of a single bet may differ from the expected value (you might win or lose), but over a large number of bets, the actual results will tend to align with the expected value.
   
   
3. Relation to House Edge
   The house edge is the negative expected value from the player’s perspective. It represents the average profit the casino earns per bet.
   For example: If the EV of a Banker bet is −0.010579, the house edge is 1.06%.

Relationship Between Expected Value and Player Strategy:

•  Choose higher expected value options: Players should select bets with an expected value close to 0 or with smaller negative values, such as Banker (EV = -0.010579).
•  Avoid low expected value options: Bets like Tie (EV = -0.143596), while offering high payouts, result in greater long-term losses.

Summary：

1. Expected value is a metric used to measure the potential long-term gain or loss from repeated bets.
2. In Baccarat:
   •  Banker has the highest expected value (smallest loss) and is the most stable betting option.
   •  Player comes next, but with a slightly higher house edge.
   •  Tie has the lowest expected value (greatest loss) and is not recommended for long-term betting.
3. Analyzing expected value allows players to make more rational betting decisions and reduce long-term losses.

Further Reading:

Probability