Re: Mr. Mechanical's Chases Baseball

Taken from wizardofodds

**<CENTER> </CENTER><CENTER> </CENTER><CENTER>The Martingale</CENTER>**
Every week I receive two or three emails asking me about the betting system by which a player doubles his/her bet after a loss. This system is generally played with an even money game such as the red/black bet in roulette or the pass/don't pass bet in craps and is known as the Martingale. The idea is that by doubling your bet after a loss, you would always win enough to cover all past losses plus one unit. For example if a player starts at $1 and loses four bets in a row, winning on the fifth, he will have lost $1+$2+$4+$8 = $15 on the four losing bets and won $16 on the fifth bet. The losses were covered and he had a profit of $1. The problem is that it is easier than you think to lose several bets in a row and run out of betting money after you've doubled it all away.

In order to prove this point I created a program that simulated two systems, the Martingale and flat betting, and applied each by betting on the pass line in craps (which has a 49.29% probability of winning). The Martingale bettor would always start with a $1 bet and start the session with $255 which is enough to cover 8 losses in a row. The flat bettor would bet $1 every time. The Martingale player would play for 100 bets, or until he couldn't cover the amount of a bet. In that case he would stop playing and leave with the money he had left. In the event his 100th bet was a loss, he would keep betting until he either won a bet or couldn't cover the next bet. The person flat betting would play 100 bets every time. I repeated this experiment for 1,000,000 sessions for both systems and tabulated the results. The graph below shows the results:

<CENTER>

</CENTER>As you can see, the flat bettor has a bell curve with a peak at a loss of $1, and never strays very far from that peak. Usually the Martingale bettor would show a profit represented by the bell curve on the far right, peaking at $51; however, on the far left we see those times when he couldn't cover a bet and walked away with a substantial loss. That happened for 19.65% of the sessions. Many believers in the Martingale mistakenly believe that the many wins will more than cover the few loses.

In this experiment the average session loss for the flat bettor was $1.12, but was $4.20 for the Martingale bettor. In both cases the ratio of money lost to money won was very close to 7/495, which is the house edge on the pass line bet in craps. This is not coincidental. No matter what system is used in the long run, this ratio will always approach the house edge. To prove this point consider the Martingale player on the pass line in craps who only desires to win $1, starts with a bet of $1, and has a bankroll of $2,047 to cover as many as 10 consecutive losses. The table below shows all possible outcomes with each probability, expected bet, and return.

<CENTER><TABLE class=contenttable cellSpacing=0 cellPadding=3 bgColor=#e0d18b border=1><TBODY><TR><TD align=middle colSpan=8>Possible outcomes of Martingale up to ten losing bets

</TD></TR><TR><TD align=middle>Number

of losses

</TD><TD align=middle>Final

outcome

</TD><TD align=middle>Highest

bet

</TD><TD align=middle>Total

bet

</TD><TD align=middle>Net

outcome

</TD><TD align=middle>Probability

</TD><TD align=middle>Expected

bet

</TD><TD align=middle>Expected

return

</TD></TR><TR><TD align=right>0

</TD><TD align=middle>Win

</TD><TD align=right>1

</TD><TD align=right>1

</TD><TD align=right>1

</TD><TD align=right>0.49292929

</TD><TD align=right>0.49292929

</TD><TD align=right>0.49292929

</TD></TR><TR><TD align=right>1

</TD><TD align=middle>Win

</TD><TD align=right>2

</TD><TD align=right>3

</TD><TD align=right>1

</TD><TD align=right>0.24995001

</TD><TD align=right>0.74985002

</TD><TD align=right>0.24995001

</TD></TR><TR><TD align=right>2

</TD><TD align=middle>Win

</TD><TD align=right>4

</TD><TD align=right>7

</TD><TD align=right>1

</TD><TD align=right>0.12674233

</TD><TD align=right>0.88719628

</TD><TD align=right>0.12674233

</TD></TR><TR><TD align=right>3

</TD><TD align=middle>Win

</TD><TD align=right>8

</TD><TD align=right>15

</TD><TD align=right>1

</TD><TD align=right>0.06426732

</TD><TD align=right>0.96400981

</TD><TD align=right>0.06426732

</TD></TR><TR><TD align=right>4

</TD><TD align=middle>Win

</TD><TD align=right>16

</TD><TD align=right>31

</TD><TD align=right>1

</TD><TD align=right>0.03258808

</TD><TD align=right>1.01023035

</TD><TD align=right>0.03258808

</TD></TR><TR><TD align=right>5

</TD><TD align=middle>Win

</TD><TD align=right>32

</TD><TD align=right>63

</TD><TD align=right>1

</TD><TD align=right>0.01652446

</TD><TD align=right>1.04104089

</TD><TD align=right>0.01652446

</TD></TR><TR><TD align=right>6

</TD><TD align=middle>Win

</TD><TD align=right>64

</TD><TD align=right>127

</TD><TD align=right>1

</TD><TD align=right>0.00837907

</TD><TD align=right>1.06414175

</TD><TD align=right>0.00837907

</TD></TR><TR><TD align=right>7

</TD><TD align=middle>Win

</TD><TD align=right>128

</TD><TD align=right>255

</TD><TD align=right>1

</TD><TD align=right>0.00424878

</TD><TD align=right>1.08343900

</TD><TD align=right>0.00424878

</TD></TR><TR><TD align=right>8

</TD><TD align=middle>Win

</TD><TD align=right>256

</TD><TD align=right>511

</TD><TD align=right>1

</TD><TD align=right>0.00215443

</TD><TD align=right>1.10091479

</TD><TD align=right>0.00215443

</TD></TR><TR><TD align=right>9

</TD><TD align=middle>Win

</TD><TD align=right>512

</TD><TD align=right>1023

</TD><TD align=right>1

</TD><TD align=right>0.00109245

</TD><TD align=right>1.11757574

</TD><TD align=right>0.00109245

</TD></TR><TR><TD align=right>10

</TD><TD align=middle>Win

</TD><TD align=right>1024

</TD><TD align=right>2047

</TD><TD align=right>1

</TD><TD align=right>0.00055395

</TD><TD align=right>1.13393379

</TD><TD align=right>0.00055395

</TD></TR><TR><TD align=right>10

</TD><TD align=middle>Loss

</TD><TD align=right>1024

</TD><TD align=right>2047

</TD><TD align=right>-2047

</TD><TD align=right>0.00056984

</TD><TD align=right>1.16646467

</TD><TD align=right>-1.16646467

</TD></TR><TR><TD align=middle colSpan=5>Total

</TD><TD align=right>1.00000000

</TD><TD align=right>11.81172639

</TD><TD align=right>-0.16703451

</TD></TR></TBODY></TABLE></CENTER>The expected bet is the product of the total bet and the probability. Likewise, the expected return is the product of the total return and the probability. The last row shows this Martingale bettor to have had an average total bet of 11.81172639 and an average loss of 0.16703451. Dividing the average loss by the average bet yields .01414141. We now divide 7 by 495 ( the house edge on the pass line) and we again get 0.01414141! This shows that the Martingale is neither better nor worse than flat betting when measured by the ratio of expected loss to expected bet. All betting systems are equal to flat betting when compared this way, as they should be. In other words, all betting systems are equally worthless.

The bottom line of this is that if you can't beat the books betting flat then the martingale system isn't going to work. If one works, the other will work, that is common sense. There is no advantage to chasing your bets. In the long run you will end up right where you would have been if you just flat betted (unless you chased too much and in that case lose more than you can afford to lose). Which makes flat betting the more preffered method because you don't have to deal with the swings and the sickly large betting amounts that could put you into retirement from gambling.