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Blackjack
Math: SD Factor
by
Henry
Tamburin
This Blackjack Math lesson 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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