Blackjack Math: SD Factor

by Henry Tamburin

This Blackjack Math lesson at the Learn to Play Blackjack Program begins with a question:
 “I was a big loser playing blackjack so I took your advice and learned the basic playing strategy. I’m still a big loser. What gives?”
Yes, I know it’s frustrating to lose, especially when you play your blackjack hands perfectly.
But this player’s experience is not uncommon. Even the most highly skilled blackjack players suffer losing streaks (been there, done that).
The reason has nothing to do with “faulty strategy,” or “the bad play of your fellow players,” or “a cheating dealer.”
No, the reason is due to what blackjack math experts call “standard deviation” (which I will refer to as SD).
If you’ve never heard of SD, don’t fret, because you are not alone. However, it is often the culprit that causes a player’s bankroll to swing wildly, so it’s important that you understand a little about SD. But I promise not to bore you with a lot of blackjack math equations. Instead, I’ll show you how SD can be used to predict and understand the results of your blackjack math.

Basically, SD is a measure of the variance (or difference) between an actual result compared to an expected result.
For example, how many heads would you expect if you flipped a coin 100 times? You probably said “50.” However, in the real world it’s rare that your outcome would be exactly 50 heads (try it, and you’ll see). Most likely, you’ll wind up with more, or less, than 50 heads, and it’s unlikely that you’ll get the same result on each 100-coin-flip trial.
If you want to know the math beforehand as to how far away you most likely will be from exactly 50 heads (i.e., the outer boundary) you need to calculate the SD.  In the case of our 100-trial coin-flip game, the math yields an SD of 5. This means that instead of ending up with exactly 50 heads as expected, you will probably end up in the range of 50 plus or minus 5 (1 SD), or between 45 and 55 heads. How probable is probable?
For a large number of trials, one SD implies that in 68.3% of the trials you will wind up between plus and minus one SD from the expected result. If you want to know the probable result with more accuracy, you can calculate twice the SD, or 2SD (95.4% certainty), or 3SD (99.7% certainty).
(Note: Our 100-coin-flip example is not a very large trial, therefore, the percent probabilities will be slightly different than the above theoretical probabilities.)
Blackjack Math: 100 COIN FLIPS
Expected result                        Possible Result       
50                                                45 to 55 range (1SD)
50                                                40 to 60 range (2SD)  
50                                                35 to 65 range (3SD)
Now let’s bet a buck on each coin flip. At the 2SD probable outcome, your result will be somewhere between 40 and 60 heads, about 95% of the time.
If heads comes up 60 times, you would be a winner of $20 (win one dollar on 60 flips and lose one dollar on 40 flips).
If instead heads came up only 40 times, you’d wind up in the red by $20.  In fact about 95% of the time you would end up winning or losing between +$20 and -$20, after 100 coin flips, and only 5% of the time would your final outcome be a win or loss outside this range.
The point is that by calculating the SD you can predict how much money you should expect to be ahead or behind in this 100-trial coin-flip game with a fair degree of certainty.
#Coin Flips     Probable at 2SD  Predicted Amount Won/Lost
      100             40 to 60 heads             +$20 to -$20
What happens if you were to wind up losing $40 after 100 coin flips? I’d look carefully at the coin, because it is highly unlikely that you would be that far outside the 2SD lower boundary of -$20 if the game were fair.  In other words, “something is rotten in the state of Denmark” (e.g., maybe someone slipped a biased, weighted coin into the game).
So let’s get back to our frustrated blackjack player. In her email she mentioned that she lost “close to $500” after three consecutive weekends of blackjack play. Let’s use SD to determine what her most likely blackjack math outcome should have been.
Our player estimated that she played 25 hours and averaged $10 per blackjack hand. We’ll assume she was dealt a standard, 100 hands per hour. This means she played 2,500 blackjack hands over the three weekends and made $25,000 worth of bets (you didn’t think it would be that much did you?). We’ll also assume that she played perfect basic blackjack strategy with a casino’s edge of about 0.5 percent.
With the above assumptions, we can calculate the blackjack math SD and determine how much money she should have won or lost with 95% certainty (i.e., 2SD).

First, let’s calculate her expected result based on the fact that even though she played perfect basic blackjack strategy, the casino still has a tiny 0.5% edge. To determine her expected result you simply multiply the total amount wagered by the casino’s edge ($25,000 x 0.5%).
In other words, her expectation was to lose $125, because the casino had the slight math edge. However, rarely will she lose exactly $125.
The calculated 2SD for this game (where the blackjack math formula is slightly different than for a coin flip) is approximately $1,100, therefore, the most likely outcome is that she will wind up winning or losing between +$975 and -$1,225. This range of results will occur 95% of the time, or in roughly 19 out of 20, 25-hour playing sessions.
2,500 BJ Hands at $10 per hand 
Expected Result         Possible Result at 2SD         
    -$125                           +$975 to -$1,225
If you compare her actual result — losing $500 — with the projected 2SD outcome of +$975 to -$1,225, you see that losing $500 is well within the expected range.
This means that her $500 loss was not at all abnormal, or “unexpected.” In fact, the blackjack math tells us that she had almost a 25% chance of losing at least $500 for her 25 hours of play.
So, for every four trips she takes, she can expect to end up $500 or more in the red once, on average. Her result, therefore, wasn’t the least bit unusual or expected.
What happens if she plays blackjack more? Will she ever get a shot at recouping her loss? It’s possible, but the chance diminishes the longer she plays. Just look at her 2SD probable outcome as she plays more hands (see below). Notice that the more hands she plays, the more the range of the probable outcome is skewed to the losing side, and eventually, at 200,000 hands, she has virtually no chance of showing a profit.
So even though luck plays a big part in your outcome in the short term, over time the casino’s edge will prevail, and you will come closer to the expected outcome, percentage wise (which in this case is a net loss).
# Hands Played   Expected Result         Outcome at 2SD
      2,500                     -$125                     +$975 to -$1,225
    10,000                     -$500                     +$1,700 to -$2,700
    25,000                     -$1,250                  +$2,230 to -$4,730
    50,000                     -$2,500                  +$2,420 to -$7420                
  100,000                     -$5,000                  +$1,960 to -$11,960
  200,000                     -$10,000                 -$160 to -$19,840
So what’s the lesson learned in all this? First, experiencing losing sessions as a basic blackjack strategy player is quite normal and should come as no surprise. The reason you have some winning and some losing blackjack sessions is due to the natural fluctuations of the game (i.e., SD). Secondly, in the short term you could experience many consecutive winning or losing sessions, because luck has a lot to do with your outcome.
Thirdly, the longer you play, the more likely your final outcome will be a net loss, because the blackjack math in the casino’s favor will ultimately prevail over “luck.”

Blackjack Math is followed by Elimination BJack
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