Gambling Streaks and the Standard Deviation, Part 3: Uneven Payoff Games

Gambling Streaks and the Standard Deviation, Part 3: Uneven Payoff Games


Gambling Winning and Losing Streaks and the
Standard Deviation
Part 3: Uneven Payoff Games
In Part 1, we saw how the deviation in the
expected return changes with the length of
play. In Part 2, we saw how this applies to
games where the house has an advantage.
Here I have a new gambling simulation. This
thin red line represents your change in bankroll
when betting on an even-money bet in roulette.
The thick red line shows the same thing for
betting on a single lucky number, which pays
35 to 1 if the ball lands on your number.
Both types of bets have the same house edge,
5.3%, but the lucky number bet has a much
larger standard deviation. I built this graph
with a zoomed-out scale so you can see the
wild results.
The lucky number bet usually loses, as shown
by this downward slope, but 1 in 38 times
on average, it wins 35 units, as shown by
the sharp jumps in the zig-zag pattern. On
average, you can expect about 10 or 11 wins
in a 400-bet session, the length of this graph.
For either type of bet, you can expect to
lose 21 bets on average in a 400-bet session,
so the average endpoint is here, 21 units
below the starting point.
These two small black lines show plus and
minus one standard deviation for the even-money
bet, whereas the big grey grid lines are spaced
at 115 units, the standard deviation for the
lucky number bet. In 2/3 of the sessions,
you can expect to end up between these two
lines for the even-money bet, or between these
two lines for the lucky number bet.
Let’s try several sessions and see what happens.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20.
You can see that a few extra lucky wins in
a session greatly increases your bankroll,
whereas being short a few expected wins ends
up with an extra-large loss for the session.
Like that.
Let’s review the summary table for the even-money
bet on Red. For a whole-day session of 400
bets, the standard deviation is 20 and the
chance of a positive day is 15%.
Here’s the summary table for the lucky number
bet. The standard deviation values are much
larger. For the whole-day session of 400 bets,
the standard deviation is 115, and the chance
of a positive result is 43% instead of 15%.
Does this mean you should bet on a lucky number
instead of an even-money bet? The answer is
yes, IF you can afford to put a lot more money
at risk. Your moderate losing session is -136
bets, versus -41 bets in the even-money game.
Remember that you have a 1-in-6 chance of
doing even worse than this moderate loss.
Certainly, though, the lucky-number bet is
more exciting to win.
If you’re interested in seeing the table for
blackjack card counting, continue with Part
4.

2 thoughts on “Gambling Streaks and the Standard Deviation, Part 3: Uneven Payoff Games”

  1. Great video… thank you for posting

    How would the results differ if the single number better say bet 1 unit for say the 1st 30 wagers than martingaled so that he always has a win through say 120 spins? Of course there will eventually be runs off misses longer than 120 but how do wins especially close to a martingale wage increase affect the winning session percentage?

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