Standard deviation in blackjack

Last Update: May 22, 2005

This appendix presents information pertinent to the standard deviation in blackjack. It is based on the following rules and total dependent basic strategy.

• Six decks
• Dealer stands on soft 17
• Double on any first two cards
• Double after split allowed
• Late surrender allowed
• Resplit aces allowed
• Player may resplit to four hands
• Cut card

This table reflects a standard deviation of 1.1418.

Here is a summary, which answers the frequently asked question, what is the probabiity of a net win, loss, and push.

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

Following is the standard deviation per individual hand, playing flat betting 1 to 3 hands at the same time. These numbers are courtesy of Marsha Ness, who used Blackjack Audit simulation software. The rules are 6 decks, dealer stands on soft 17, double any two cards, double after split, no surrender, split up to 4 hands, no resplitting aces, no drawing to split aces.

For example, the total standard deviation of three hands of $100 each, played at the same time, would be $100 × sqr(3) × 1.51957 = $263.20. By comparison, the standard deviation of a single hand of $300 would be $300 × 1.15514 = $346.54.

According to Professional Blackjack by Stanford Wong (page 203), the variance for similar rules is 1.32 and the covariance is 0.48. The total variance of n hands would be 1.32*n + 0.48*n*(n-1). Take the final square root to get the standard deviation.

The next table is a practical application of the standard deviation. It is useful if you wish to know the probability of a large net loss or win after a session of flat betting. The left column represents the number of hands in the session. The top row represents the probability that the result, after adjusting for the house edge, will exceed the table value. The body of the table represents the number of units won or lost, after adjusting for the house edge.

For example suppose a blackjack player loses 100 units over a session of 1000 bets. Assuming an 0.4% house edge, 4 of the losses are expected due to the house edge and 96 are the result of bad luck. The player wishes to know the probability of a loss of this magnitude. The table shows the probability of a loss of 95 units to be 0.5%. Thus the player can expect to lose 95 units or more about 1 session in 200.