Gambling Streaks and the Standard Deviation, Part 2: Unfair Games

Gambling Streaks and the Standard Deviation, Part 2: Unfair Games


Gambling Winning and Losing Streaks and the
Standard Deviation
Part 2: Unfair Games
In Part 1, we saw how the deviation in the
expected return changes with the length of
play. This concept becomes important when
we move from a fair coin-flip game to an unfair
casino game.
Here I’ve set up another simulation with two
gambling games plotted on the same graph.
This black line shows the change in bankroll
for a fair coin-flip game, while this red
line shows the same thing for an unfair game,
betting on Red at roulette. For a standard
American two-zero wheel, the chance of winning
is 18/38 and the house edge is 5.3%.
I triggered the win-lose decisions on the
same sequence of random numbers, so the fair
and unfair game results are correlated. The
win/loss decision for the two games is usually
the same, but randomly, about 1/38 of the
time, the roulette bet loses when the coin-flip
bet wins. So after 400 bets, the roulette
game is behind the fair game by about 21 bets.
These horizontal lines are spaced at 20 bets,
which is the standard deviation of the win
or loss for a session of 400 bets. I’ll press
the key several times.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
You can see that the roulette results fall
behind the fair game by about 21 bets in each
session. So you need about 1 standard deviation
of positive good luck just to overcome the
house edge and break even.
What about sessions shorter or longer than
400 bets? Let’s take another look at the summary
table for the fair coin-flip bet first. The
chance of a positive result is 50% no matter
how long you play. The unlucky and lucky session
results are centered on the average final
return, zero.
Here’s the table for the roulette even-money
bet. The number of plays in this column represent
one single spin, and roughly one hour, one
day, and one week of playing roulette, one
bet per spin. The standard deviations are
the same as before.
For a single spin, the chance of a positive
result is 47 percent, a nearly even game.
For a one-hour session, the average loss is
3 bets. An unlucky session, one standard deviation
down, is a loss of 10; and a lucky session,
one standard deviation up, is a win of 4.
The chance of a positive end result is still
decent, 35%.
For a one-day session, the average loss is
21 bets, and even at +1 standard deviation,
you’re just barely breaking even. This is
what we saw in the simulation. The chance
of a positive result is 15%.
For a one-week session, you can expect to
lose 130 bets on average, and you can expect
to win 47% of the bets you make, plus or minus
1%. To come out ahead at the end of the week,
you need good luck at the +2.6 standard deviation
level, which will happen a little less than
one-half of 1 percent of the 1-week sessions
that you play.
Now let’s take a look at a better bet, the
Pass bet in craps. The chance of winning a
single bet is 49 percent, which is close to
an even game.
For a one-hour session, the average loss is
1 bet, and the unlucky and lucky end results
are almost equal. Your chance of coming out
ahead is quite good, 46%
For a one-day session, the average loss is
6 bets, and the chance of winning is still
a decent 39%.
For a one-week session, you can expect to
lose 35 bets on average, and your chance of
ending up ahead is 24%, not great but certainly
better than roulette.
These good results apply only to the Pass
and Don’t Pass even-money bets. Many craps
bets have a house edge as bad as roulette
or even worse. However, there are some craps
bets with a house edge of exactly zero, which
is the subject of another video.
In summary, the shorter you play, the better
your chance of winning, and conversely, the
longer you play, the more certain it is that
you will lose. Playing a game with a smaller
house edge helps tremendously, especially
if you play a long time.
So far we’ve looked only at even-payoff games.
What about an uneven payoff game, like betting
on a lucky number at roulette, which pays
35 to 1? That’s the subject of Part 3.

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