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Originally published in Bluff Europe magazine, July, 2011 SEQUENCES
by Les Golden
While
we’re waiting for the U.S.
government to figure out how to tax winnings on internet gambling,
and soon thereafter declaring that poker really is a game of skill
and so all the poker websites are free to open as many rooms as they
like, let’s get back to the Small Martingale. It’s a part of
what’s been known for hundreds of years as “gambler’s ruin.” Gambler’s
ruin refers to several
concepts, going back to one of my personal heroes, Christian Huygens,
an early believer in the possibility of extraterrestrial life and
perhaps the first theoretical physicist, among other things. Today,
gambler’s ruin most commonly refers to the inevitability of a
gambler losing in the “long run” in a casino game in which the
house has an advantage despite any systems they employ. As I said
last time, we also require that the probabilities of the game don’t
change as the game is played and that the trials are independent. In short,
the gambler will eventually
go broke in a game favoring the house, regardless of the betting
system. It applies to craps, unbiased roulette, the wheel of
fortune, keno, and the state-run, and therefore legal, numbers racket
known as the lottery which generates so much money for bottom-feeding
government bureaucrats. These are the ones who tell us, “Poker is
a game of chance with no skill involved.” Gamblers who think they can consistently win at any game other than poker, blackjack, and baccarat ignore this mathematical fact. So, as a service to you Bluffers, I’m trying to show how one popular particular system, the Small Martingale, is bound to fail. I’m using flips of a coin and the roulette wheel to provide the basic arguments.
MY FAVORITE THREE WAYS TO DO IT
I can see three ways to show this inevitability. One way is to calculate the probability of experiencing a certain number of losses in a row after a given number of trials. In the context of the roulette wheel, in 73 spins, for example, you can fairly easily use some probability calculations to show that a 50.3% chance exists that you’ll lose at least 6 spins in a row. Similarly, a 77.2% chance exists that you’ll lose at least 6 spins in a row if you spin the wheel 150 spins, and more than a 90% chance exists that you’ll lose at least 6 spins in a row if you spin the wheel 250 times. I don’t find this way of demonstrating the inevitability very persuasive. It’s not scary enough. A second way to illustrate gambler’s ruin is to show the average number of trials until a given sequence of losses occurs. Over the years, various mathematicians have independently shown, in a much more difficult analysis, that a simple, cool formula yields the result. In the context of coin flips, a sequence of 4 heads (or tails) appears on the average after 30 flips, a sequence of 5 heads appears on the average after 62 flips, and a sequence of 6 heads appears on the average after 126 flips. As a hint to the math geeks to the nature of the general formula, note that 62 = 30 + 32 and that 126 = 62 + 64. The
general formula for
the average number of trials necessary until a sequence of length The first figure displays the formula in terms of flips of a coin, heads beings considered a “loss.” Either heads or tails give the same result, of course. As expected, most of the time a loss is followed by a win and the length of the sequence is one. A large number of times two losses occur in a row. As the number of losses in the sequence increases, the frequency with which that sequence appears decreases quickly. Long sequences of losses in fact do occur. In the 10,000 flips, 252 sequences of 10 losses in a row, 124 sequences of 11 losses in a row, and 82 sequences of 12 losses in a row occurred on the average. The appearance of such large number of losses in a row is an important result when you’re worried about losing your entire bankroll. Now, that is a cool result, that the average number of trials is always an integer, but it is only an average. Gamblers need to have an idea of the spread in the number of flips needed before a sequence appears. Igor and I have let the computer run Monte Carlo simulations and we find, discouragingly, that the spread around the average is often comparable to the value of the actual average. Telling a roulette player that he might go bust in 64 plus or minus 50 spins isn’t very illuminating.
ANOTHER KIND OF SEQUENCE
I prefer a
third means of expressing
the gambler’s ruin. Flip the coin 100 times. How many times in
those flips will you get a sequence of heads (or tails) a given
number of times? In roulette talk, we want to determine the number
of times in 100 spins of the wheel you’d get a given number of
losses. I choose 100 because that’s about the number of spins of
the wheel in an hour or so, depending on the total number of players. Now, that
would be a valuable result. It would tell us, for example, what our
minimum bet should be. In
the table I presented last time, we see that with a table limit of
$500, the Small Martingale system would not allow you to recoup your
losses if you were betting $25 and had six losses in a row. On the
other hand, if you were betting $1, you could lose ten times in a row
before not being able to recoup your losses. It would also tell us what kind of a bankroll you would have to bring. If, for example, you’re betting $5, and brought along $1000, you could go bankrupt if you lost eight times in a row twice.
To analyze this approach, Igor and I ran a second Monte Carlo simulation. First, we did it on a 50-50 game, a “fair” game, in which the house has no edge. Igor lets the computer randomly pick a number between 0 and 1. If the number is less than 0.50, the trial (here, a flip of a coin) is considered a loss (or, to be game-specific, a heads); if the number is greater than or equal to 0.50, the trial is considered a win. We keep track of the number of times various numbers of wins and losses occur in a row. We
perform 10,000 trials
and I do it 50 times. The number of each type of sequence is then
divided by 100 to provide the expected number of sequences in 100
trials. We also let the computer calculate the spread in the values
among the 50 experiments. We then determine the number of times 1,
2, 3, 4, 5, and so on losses in a row appear. The results are shown
in the second figure, for 3 through 12 losses (“heads”) in a row,
with the spread in values indicated by the vertical lines next to the
data points.
Now, although one of the
meanings of the term gambler’s ruin refers to a fair game, that’s
hardly of interest to gamblers. More relevant to the casino, Igor
and I redid the simulation with the house having edges corresponding
to random numbers between 0 and 0.51, 0 and 0.52, and so on being
considered a loss. In this way, we can determine the numbers of
sequences of losses of given lengths for any game with any edge to
the house. In roulette, for example, the house has an advantage of 5.26% in all wagers except the five-number bet, where the house advantage is 7.89%. The analysis will provide the number of sequences of losses of a given length given those various house advantages. Similarly, we can provide those values for the various bets in craps.
Igor tells me he’s late for lunch, and you can get just so much nourishment from doing neat gambling calculations. So we’ll deal with the gambler’s ruin and Small Martingale systems as they relate to real games, in which the house has an advantage, next time.
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