Bad Beats in Blackjack
Bad Luck, or a Fixed Game?
By Dan Pronovost
Dan Pronovost is the owner and president ofDeepNet Technologies, makers of a wide range of advantage gambling training products and software (blackjack, poker, craps). Their web site is: www.DeepnetTech.com, and all products are available for free trial download. Dan is also the creator of the easy-to-use card counting system Speed Count, taught in the Golden Touch Blackjack courses for two years and now available in the Frank Scoblete's new book, "Golden Touch Blackjack Revolution!": www.GoldenTouchBlackjack.com/scbook.shtml.
How Unlucky is Unlucky in Blackjack?
As the owner of a company that produces advantage gambling training products and software, I often get questions that run along these lines
"I got dealt such-and-such hands in blackjack
they must be cheating!
What are the odds of this happening?"
The unhappy blackjack player just got a bad beat, and is sure the casino or dealer, be it land-based or online, is pulling a fast one. Surely there's a better chance of lightning striking them than the terrible run of cards and hands they got, which no doubt proves that the jig is up and cheaters revealed! Can Dan Pronovost, gambling math-geek extraordinaire, help champion the cause for good, and prove the on-going fraud hidden under the covers?
These kinds of questions intrigue me, not so much because of the possibility of fraud, but because the innocent questions often reveal some surprisingly tricky blackjack math problems (which, I admit, do in fact turn my crank). Having said this, a while back I was contacted with such a case, and did in fact reveal that a particular online casino was very statistically suspect of cheating by 'dealing seconds':click here if you're curious (I wasn't the only mathematician who studied this - see article for details). But I digress back to the mystery of the Bad Blackjack Beat
Bad Beat Mystery #1
A recent customer contacted my with the following Bad Beat story:
"I was playing at an online casino single deck game, excellent rules. I went down 150 bet units in the minus, flat betting, after 12 hours of play, playing about 8 hands a minute. Is it possible to lose this much: never happened to me in 20 years?"
This is a fairly standard bankroll risk-of-ruin blackjack question (ROR): what is the probability of having a final loss of 150 units (or less) in a 12 hour playing session in the noted blackjack game? There are a couple of ways to attack this problem, which all start with a good computer blackjack simulation. For this, I used my company'sblackjack training software simulator, Blackjack Audit.
I simulated the following game and rules: one deck, noDAS, H17, re-deal after every round, heads up play, flat betting. Played with perfect matching basic strategy, this game only gives the house about a 0.15% edge over the player. And in 12 hours, at 8 hands a minute, this player is going to play 5,760 hands.
For my first approach, I simulated the above game, and then used the 'ROR Simulator' inBlackjack Audit. This runs multiple blackjack 'sessions', with different bankroll and round limits. For this test, each ROR simulator session consisted of exactly 5,760 rounds of blackjack, with no lower or upper limit (i.e. the player had no upper goal or bankroll limit). The final results showed that average ending bankroll was 11 bet units in the hole, and that the standard deviation of winnings per session was about 88 bet units. This funky term is a statistical measure of the variability of the ending bankroll from session to session. The actual player lost 150 bet units, so (150 - 11)/88 = 1.58 standard deviations away from the expected ending bankroll. Dusting off the stats book on the shelf, a quick table lookup reveals the probability of this happening: 1 - 0.94 = 6%. In other words, the probability of this player ending up with a loss of 150 bet units, or more, is 6%.
This is uncommon, but not so rare that we should start a law suit. This player, unfortunately, just got a bad beat, as a result of bad luck. Had the probability been far less than 1%, that definitely would have warranted further exploration.
Another way to solve this problem is to start with the standard deviation per round for the base game (not sessions). This bypasses the ROR session simulator completely, using only the simulation for the base game. The simulation report tells us that the standard deviation per round for this game is 1.14737 bet units. Our trusty stats book tells us that the standard deviation of a sum of 'n' random events is: sd * sqrt(n), so: 1.14737 * sqrt(5760) = 87.1 bet units very close to our session result. Our average earnings per round was 0.005076 bet units, so the expected final bankroll after 5760 rounds is: 8.7 bet units. A bit off from the session result of 11 bet units, but we only ran 10,000 sessions in the ROR simulator, which leaves room for error. Using the same approach as before with these values, we get (150 - 8.7)/87.1 = 1.62 standard deviations or 5.4%, which validates the previous result.
Astute readers might ask, "Hey what if this player only had a bankroll of 150 bet units, and simply stopped after 12 hours when he hit his limit? Does this change things?" In other words: what are the odds of reaching a lower bound loss of 150 bet units in 12 hours or less? This is slightly different, and is known as trip bankroll. We can model this again in the Blackjack Risk of Ruin simulator, by placing a lower bound of 150 bet units (after which the session ends immediately) as well as the rounds limit of 5,760. Things are easier in this case, since we simply divide the number of sessions where we hit the lower limit by the total number of sessions we ran: 1095 / 10,000 = 10.95%. So, it's even more likely to be due to chance in this case.
We can also use the ROR calculators inBlackjack Audit as an alternative approach to solve this problem without the ROR simulator. There are equations that closely approximate the trip risk of ruin, based on the game's standard deviation per round and average win rate. Plugging these in, we get a trip ROR of 10.05%, which validates the ROR simulator result.
Bad Beat Mystery #2
Another player wrote me this:
"I just played 30 hands of Blackjack online using perfect basic strategy and won only 5 hands. I'm gonna put them in the Guinness book of records! In 20 years of playing Blackjack I've never had such a loss!"
This is a different kind of statistical question, called a Bernoulli trial: what is the probability of some independent event, with single probability 'p', occurring 'n' or less times?
In a typical blackjack game using perfect basic strategy, the probability of winning a hand (including blackjacks but not including ties), is roughly 45%. This is the probability 'p' above. This probability does not vary very much with different rules or games. This is generally known, but I got the figure from aBlackjack Audit simulation once again.
There are complicated approximation equations for Bernoulli trials, but for small numbers of trials like this it's easier to simply compute the exact value using the base sum-formula in a spreadsheet or program. Thankfully, ourSmart Craps simulator is chock full of Bernoulli equation modeling, since it's the basis of the method used to determine whether a dice controller's results indicate true skill. I'll save you the boring details this time, and just tell you that the probability of winning 5 or less hands of blackjack in 30 consecutive hands is 0.1090%, or 1 in a thousand (download the free trial software if you're really curious). This is very low, but not so low as to justify calling the cops, in my opinion. The problem is that 30 'trials' is not very many, and it's going to happen to one in every thousand poor souls from time to time. But its worth further exploration, and if it persisted consistently, then I'd be worried!
Bad beats are going to happen, and even the rarest events are going to occur given the huge number of games being played at any given time. Determining the difference between a bad beat and cheating is a tricky business, but possible. Any serious advantage blackjack player should use a good blackjack simulator on a regular basis, especially as they encounter different games, rules, and conditions. A good blackjack simulator and calculator are essential tools once you have honed your skills with good training software. All of our blackjack, craps and poker training products are free to download and try out, and we have software for Windows, Palm OS, and Pocket PC. Visit our web site for details:www.deepnettech.com.