applying math to blackjack

With the 6:5 blackjack payoff, the EV for accepting even money is about 20% higher than declining it ($.17 / $.83), therefore the player should accept it when it’s offered. (This option is rare since casinos are not stupid.)

The EV for accepting even money with a 6:5 blackjack payoff is 20% more.

Note that the payoff and the bet in these tables is stated as a “betting unit,” which is a way of expressing the monetary value of the bet and payoff without referring to a specific amount. Multiplying betting units by the actual bet gives the monetary results.

Also note that the odds for even money are calculated based on 15 (rather than 16) 10-point cards and 49 (rather than 52) other cards left in the deck. This is because at least one 10-point card is in the player’s hand (for him to have blackjack) and there are at least three cards on the table (assuming only the player and the dealer are playing). The calculation changes slightly with more decks and with other player’s cards on the table but not enough to swing the results in another direction.

Most players take even money claiming that “it’s foolish to risk a push and receive nothing for their blackjack.” They are convinced it’s always the right thing to do. Players who refuse even money, as they should with the 3:2 game, are therefore open to criticism as their refusal seems like a rookie mistake. The best tactic when this happens is to smile and nod in the like you just fell off the turnip truck. Trying to convince a player that he’s wrong in the middle of a blackjack game is futile.

Calculating the Profit and Loss Range

Luck is unpredictable; however, it is possible to use a formula that employs “the standard deviation for blackjack” to calculate the probability of a profit or loss. The formula for maximum loss is:

Maximum loss (with 95% confidence) =

[(Standard deviation x 2 {for 95%}) x
(Square root of the number of hands played) x
(Average bet)] -
[((Average bet) x (Number of hands played) x (House edge)) +
((Average bet) x (Number of hands played) x (Basic strategy handicap))]

The formula for maximum profits is the same except the amount attributable to the house edge and the player’s basic strategy handicap is subtracted rather than added.

Let’s use some real numbers to show how it works:

Standard Deviation = 1.17
Number of standard deviations for 95% confidence = 2
Hands Played per Hour = 55
Number of Hours of Play = 3
Average Bet = $20
House Edge = 0.5%
Basic Strategy Handicap = 1.5%

Plugging these into the formulas for maximum loss and maximum profit, we get:

Maximum Loss = $667
Maximum Profit = $535

In other words, a player betting an average of $20, at a table that plays 55 hands per hour, for three hours, where the house edge is 0.5%, and his basic strategy handicap is 1.5% has a 95% probability of losing no more than $667 and winning no more than $535.

In addition, we can also say with 95% confidence that if this player brings a bankroll of $667 to the table, he will NOT go broke in this 3-hour game.

Luck and the House Edge

Many players believe that the house edge and not luck is primarily responsible for their losses, that somehow the casino is manipulating the game to win. This is not true on two counts.

First, that part of the house edge that’s due to the game’s rules is typically 0.5%. Even when the rules are heavily in the casino’s favor, such as with a 6:5 blackjack payoff, which adds about 1.39% to the house edge, the total advantage is still less than 2% or $20 per $1,000 bet. If the $20 bettor’s P/L Range is $655 to $523, a $20 loss due to the house edge is insignificant.

Second, the house advantage due to imperfect basic strategy play can range from 0% to 10% or more, depending on the player’s basic strategy skill. The typical penalty for imperfect play is about 2% among experienced players. Even if this typical amount is combined with a 2% house advantage due to the rules, the average loss is still only 4% or $40 per $1,000 bet compared to the $667 loss that’s due to luck.

The numbers that support the statement that “the casino always wins because they have the odds advantage” just don’t add up.

The Law of Large Numbers

Most of what’s been said so far about luck applies to the short term (for example, a single blackjack session, one evening of blackjack play). What can the player expect in the long term? Here, it’s almost all good news. Like most things in life, if you wait long enough, things even out. So it is with blackjack — over a long period, wins and losses due to luck will tend to cancel each other out, and what remains are only the losses resulting from the house edge and imperfect blackjack play.

 The more you play, the more you will approach breakeven.

It’s really just common sense. No one has bad or good luck forever. Statisticians call this effect the Law of Large Numbers which is a fancy way to say, “things average out over time.” The law states “that as a sample size grows, the mean gets closer to the average of the whole population.” For blackjack players, the “sample” is the blackjack session, the “population” is the complete history of the player’s wins and losses, and the “expected average of the population” is the average of all wins and losses less the sum of all losses due to the house edge.

The following diagram illustrates this. The blue bars represent the imagined profits or losses of a succession of blackjack sessions. The green line is the average of all profits and losses. The red line is cumulative loss due to the house edge.” Over time, the red line and the green line will converge.