Variance in Blackjack

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.

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.

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)

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.

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.

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.

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 Side Bets
• Introduction to Card Counting
• The Hi-Lo Count
• The Wizard's Ace-Five Count
• When to Surrender in Blackjack
• Value of a Free Ace
• Dealer Exposes Both Cards
• Splitting Strategy for the Back Players
• Double after Splitting Aces Allowed
• Continuous Shuffling Machines
• Cut Card Effect
• Variance in Blackjack
• 678 and 777 Bonuses in Blackjack
• Risk of Ruin in blackjack
• Total Dependent vs. Composition Dependent Basic
• Expected Returns with Infinite Decks
• Dealer Odds in Blackjack under U.S. Rules
• Dealer Odds in Blackjack under European Rules
• Effect of Card Removal in Blackjack
• Blackjack Expected Values
• Estimating Number of Decks in Online Blackjack
• Composition-Dependent Strategy for Single Deck and Dealer Stands on Soft 17
• Composition-Dependent Strategy for Single Deck and Dealer Hits on Soft 17
• Composition-Dependent Strategy for Double Deck and Dealer Stands on Soft 17

Written by: