Mythological Moment: Progression systems – part 2

Copyright 2001, The Mayor

When Einstein came up with his famous E = mc^{2} he did not suddenly invent a new law that didn’t exist previously. He used mathematics to rigorously derive the equation, and left it to a future generation to realize its potential applications. Mathematics helps us model reality, and can be used to make reliable predictions about the effects of certain actions we take.

Other not so famous mathematicians long ago arrived at their own theorems in statistics and probability theory, which bear directly on money management and betting schemes. Here are the basic results:

- A betting scheme cannot change the house edge.
- All betting systems lead to the same result: the house will win at a long term rate of (house edge) ´ (total dollars wagered).
- Betting systems have enormous variances, leading to higher probabilities of ruin.

To put this to the test, I wrote a computer program to model a very simple game. In it, a coin is tossed repeatedly. If the coin comes up heads you win, tails you lose. Begin by betting $1, and double your bet until you win. After a win, begin over again by betting $1. Thus, if the sequence is tails, tails, tails, heads, you will have bet $1, $2, $4, and $8. You won your final $8 bet, but you lost the previous $1 + $2 + $4 = $7. Thus, you are ahead $1.

What happens if we toss the coin exactly 20 times? I’ll call that a "session." Here are the results of a typical session:

How many tosses: 20

High point: $7 at toss 20

Low point : -$250 at toss 16

Final total: $7

I then ran this program for one hundred million sessions, consisting of 20 tosses each, and accumulated the statistics. Here they are.

Average high point during session: $10.00

Average low point during session: $-83.68

Maximum high point for all sessions: $20.00

Maximum low point for all sessions: $-2,097,151

Number of winning sessions: 86,188,733

Number of losing sessions: 13,811,267

Average win at a winning session: $7.39

Average loss at a losing session: $46.12