HOW I TRIED TO BEAT VEGA$An Exercise in the Right and Wrong Ways
to Use Probability
Dr. Shane RedmondSeptember 23, 2013
Disclaimer:
No EKU funds were used at casinosto research this talk.
This talk is also not an endorsement ofgambling.
The game: Roulette
Invention has been attributed to mathematiciansBlaise Pascal, Don Pasquale, and many others.
The wheel: 38 numbered slots
18 red18 black0 and 00 are green in American roulette(European roulette has only one 0, 37 slots)
The bets: Bet on one individual number
Payout is 35 to 1Meaning if you bet $1 and win, you will be given$36, for a net gain of $35.P(lose) = 37/38 =.9737
The bets: Corner bets
Payout is 8 to 1
P(lose) = 34/38 = .8947
Outside bets: Columns
Payout is 2 to 1
P(lose) = 26/38 = .6842
Outside bets: Color, parity, high/low
Payout is 1 to 1
P(lose) = 20/38 = .5263
So our odds are best with the outside bets
Let’s consider playing against a streak.For example, if we bet red, let’s bet red againbecause the probability of not seeing red twicein a row is (20/38)*(20/38) = .2770.
In fact, the probabilities of not seeing redSTREAK PROBABILITY
1 .5263
2 .2770
3 .1458
4 .0767
5 .0404
6 .0213
7 .0112
8 .0059
9 .0031
10 .0016
11 .0009
( 2038 )𝑛
Gambler’s FallacyJust because the wheel came up black last time, it does NOT make red MORE likely to come upthe next time. The probability of red is the SAMEon each spin, no matter the previous outcomes.
This is called the “Gambler’s Fallacy” or “MonteCarlo Fallacy” after one crazy night in 1913where black came up on the roulette wheel 26times in a row.
However, we’re not falling for the Monte Carlo Fallacy
Flip a coin. P(heads) = ½
Flip a coin twice, P(heads twice) = ¼
So how do we take advantage of these favorable probabilities?
If we bet $1, lose, and bet $1 again, we do not come out ahead.But, if we double the second bet to $2, we do come out ahead:-$ 1 - $ 2 + 2($ 2) = $ 1Loss onBet #1
Amount ofBet #2
PayoutBet #2
Overallprofit
Similarly, if we lose twice, double the thirdbet again to cover previous losses andensure a profit.
Loss onBet #1
Loss onBet #2
PayoutBet #3
Overallprofit
-$ 1 - $ 2 - $ 4 + 2($ 4) = $ 1
Amount ofBet #3
We can continue this process of doubling our bets to cover previous losses for streaksof any length.
This method is an example of a Martingale System.
First gained popularity in 18th century France.
First studied mathematically byPaul Levy (1934).
Named “martingale” by Jean Ville (1939).
Table of bets to make betting against a streakSTREAK PROBABILITY BET
1 .5263 $ 1
2 .2770 $ 2
3 .1458 $ 4
4 .0767 $ 8
5 .0404 $ 16
6 .0213 $ 32
7 .0112 $ 64
8 .0059 $ 128
9 .0031 $ 256
10 .0016 $ 512
11 .0009 $ 1024
2𝑛− 1
What problems do you see with ourMartingale System?
STREAK PROBABILITY BET TOTAL BET PAID
1 .5263 $ 1 $ 1
2 .2770 $ 2 $ 3
3 .1458 $ 4 $ 7
4 .0767 $ 8 $ 15
5 .0404 $ 16 $ 31
6 .0213 $ 32 $ 63
7 .0112 $ 64 $ 127
8 .0059 $ 128 $ 255
9 .0031 $ 256 $ 511
10 .0016 $ 512 $ 1023
11 .0009 $ 1024 $ 2047
-1
Sum ofprevious bets, or
Practical problems: You need a large initial bankroll.
STREAK PROBABILITY BET TOTAL BET PAID
1 .5263 $ 1 $ 1
2 .2770 $ 2 $ 3
3 .1458 $ 4 $ 7
4 .0767 $ 8 $ 15
5 .0404 $ 16 $ 31
6 .0213 $ 32 $ 63
7 .0112 $ 64 $ 127
8 .0059 $ 128 $ 255
9 .0031 $ 256 $ 511
10 .0016 $ 512 $ 1023
11 .0009 $ 1024 $ 2047
In all cases,the profitis only $ 1.
Practical problems: The payout is small.
The Martingale System is like a “reverse lottery”
Lottery• We know the odds of losing
are high.• Profit when we win:
great (millions of $$)• Consequences of losing:
small ($1-$2)
Martingale system• We think the odds of losing
become low.• Profit when we win:
small (usually the initial bet)• Consequences of losing:
great (may hundreds times the initial bet)
Some ways around the small profit
*Start with bets larger than $ 1(Of course, this means a much bigger initial bankroll is necessary.)*Run several bets at once(For example, bet red and odd and high numbers,continuing the doubling only for the bets that lose)*Use other patterns other than doubling(For example, $ 1 -> $ 2 -> $ 5 -> $ 10 -> $ 20 -> $ 50 -> $ 100 -> $ 200 -> $400 …)
Practical problems: Complementary Drinks
The Hangover, 2009, Warner Bros.
Practical problems: Table Limits
Classic sign
Modern digital sign
Practical problems: Pit Boss
1995 – Universal Pictures
Practical problems: Pit Boss
$ 400 bet on red
represented a risk of$ 1 + $ 2 + $ 5 + $ 10+ $ 20 + $ 50 + $ 100 + $ 200 + $400 = $788
for a profit of $ 12
Mathematical problems: The probability is never zero …
because we do not have an infinite amount of time to play the game or an infinite budget.
Mathematical problems
*The probability of a streak occurring is not thebest way to look at the outcomes.
*Long streaks are not as uncommon as theprobabilities suggest.
Expected Value
EV = (profit from win) × (probability of win) + (amount of loss) × (probability of loss)
The expected value is the value one would“expect” to find if one could repeat a randomprocess over an very large number of trials and take the average of the values obtained.
Expected Value: one play
EV = (profit from win) × (probability of win) + (amount of loss) × (probability of loss)
Expected Value: streak of n plays
($1 )(1−( 2038 )𝑛)+( 2038 )
𝑛
(−$ (2𝑛−1 ) )
Probability of win before a streak of n losses
Probability of streak of n losses
Amount of loss
¿1−( 2038 )𝑛
−2𝑛( 2038 )𝑛
+( 2038 )𝑛
=1−( 4038 )𝑛
=𝟏−(𝟐𝟎𝟏𝟗 )𝒏
STREAK PROBABILITY BET TOTAL BET PAID
EXPECTED VALUE
1 .5263 $ 1 $ 1 - $ .05
2 .2770 $ 2 $ 3 - $ .11
3 .1458 $ 4 $ 7 - $ .17
4 .0767 $ 8 $ 15 - $ .23
5 .0404 $ 16 $ 31 - $ .29
6 .0213 $ 32 $ 63 - $ .36
7 .0112 $ 64 $ 127 - $ .43
8 .0059 $ 128 $ 255 - $ .51
9 .0031 $ 256 $ 511 - $ .59
10 .0016 $ 512 $ 1023 - $ .67
11 .0009 $ 1024 $ 2047 - $ .76
Expected Value: 1:1 outside bets
Outside bets: Columns
Payout is 2 to 1
P(lose) = 26/38 = .6842
STREAK PROBABILITY BET TOTAL BET PAID
PROFIT EXPECTED VALUE
1 .6842 $ 1 $ 1 $ 2 - $ .05
2 .4681 $ 1 $ 2 $ 1 - $ .09
3 .3203 $ 2 $ 4 $ 2 - $ .14
4 .2192 $ 3 $ 7 $ 2 - $ .19
5 .1500 $ 4 $ 11 $ 1 - $ .23
6 .1026 $ 6 $ 17 $ 1 - $ .28
7 .0702 $ 9 $ 26 $ 1 - $ .33
8 .0480 $ 14 $ 40 $ 2 - $ .38
9 .0329 $ 21 $ 61 $ 2 - $ .44
10 .0225 $ 31 $ 92 $ 1 - $ .49
11 .0154 $ 47 $ 139 $ 2 - $ .54
12 .0105 $ 70 $ 209 $ 1 - $ .60
13 .0072 $ 105 $ 314 $ 1 - $ .66
14 .0049 $ 158 $ 472 $ 2 - $ .72
15 .0034 $ 237 $ 709 $ 2 - $ .78
2:1outsidebets
Expected Value: a recursive formula
𝑒𝑛=𝑒𝑛−1+𝑝𝑛−1(𝑤𝑛 (1−𝑝 )+𝑏𝑛−1−𝑏𝑛𝑝 )
= Expected value for spin n
= Expected value for spin n - 1
= probability of a loss on this spin
= profit for win on spin n
= Sum of betsfor spin n
= Sum of betsfor spin n -1
STREAK PROBABILITY BET TOTAL BET PAID
PROFIT EXPECTED VALUE
1 .8947 $ 1 $ 1 $ 8 - $ .05
2 .8006 $ 1 $ 2 $ 7 - $ .10
3 .7163 $ 1 $ 3 $ 6 - $ .14
4 .6409 $ 1 $ 4 $ 5 - $ .18
5 .5734 $ 1 $ 5 $ 4 - $ .21
6 .5131 $ 1 $ 6 $ 3 - $ .24
7 .4591 $ 1 $ 7 $ 2 - $ .27
8 .4107 $ 1 $ 8 $ 1 - $ .30
9 .3675 $ 2 $ 10 $ 8 - $ .34
… … … … … …
39 .0131 $ 57 $ 510 $ 3 - $ 1.54
8:1 corner bets
STREAK PROBABILITY BET TOTAL BET PAID
PROFIT EXPECTED VALUE
1 .9737 $ 1 $ 1 $ 35 - $ .05
2 .9481 $ 1 $ 2 $ 34 - $ .10
3 .9231 $ 1 $ 3 $ 33 - $ .15
… … … … … …
35 .3932 $ 1 $ 35 $ 1 - $ 1.21
36 .3829 $ 2 $ 37 $ 35 - $ 1.26
… … … … … …
53 .2433 $ 2 $ 71 $ 1 - $ 1.81
54 .2369 $ 3 $ 74 $ 34 - $ 1.85
… … … … … …
… … … … … …
116 .0453 $ 15 $ 508 $ 32 - $ 3.97
35:1singlenumberbets
Are streaks that rare?
We calculated earlier with a $ 511 bankroll, you could bet on red/black for a streak of up to 9 spins with a probability of not winning in those 9 spins of
=.0031
But how likely is it to hit a streak of 9 or more andlosing our bankroll before we earn back our $ 511?
Likelihood of hitting a bad streakSo, our chances of not bankrupting (not hitting a streakof 9 or more against us) on the first 9 spins is 1 - .0031 = .9971.The probability of going bankrupt on each subsequent spin is
So, the probability of not going bankrupt after n > 9spins is
.9971
Likelihood of staying solvent after n spins
Number of spins Probability Not Bankrupt
9 .9971100 .8723200 .7531300 .6502400 .5614500 .4847750 .33571000 .2325
Lessons learned
Thanks!!!
and …
This Saturday 9/28
Bluegrass Undergraduate Mathematics Symposium(B.U.M.S.)
Centre College in Danville
FREE to undergraduates if register online bySept. 24 (tomorrow!)
Grad students – contact me for FREE registrationFREE LUNCH TO ALL WHO PRE-REGISTER!
http://web.centre.edu/bums/Home.html