The Reasons Why We Sometime Still Lose (Part 2)

“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?”

In Part 1 we looked at how standard deviation can explain the ups and downs of our bankroll. Part 2 dwells further into the important concept of standard deviation.

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 play. Let’s use SD to determine what her most likely outcome should have been. 

Our player estimated that she played 25 hours of blackjack and averaged $10 per hand. We’ll assume she was dealt a standard, 100 hands per hour. This means she played 2,500 hands of blackjack 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 strategy with a casino’s edge of about 0.5 percent. 

With the above assumptions, we can calculate the 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 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 mathematical edge. However, rarely will she lose exactly $125. The calculated 2SD for this blackjack game (where the 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 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 more blackjack? 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 -$7,420               

  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- strategy player is quite normal and should come as no surprise. The reason you have some winning and some losing 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 math in the casino’s favor will ultimately prevail over “luck.”

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