May 07, 2003
The Gambler's Ruin

Now that we all know more than we ever thought we'd know about Bill Bennett's love of slot machines, let's talk numbers. What are the odds that you can actually break even or come out ahead when playing the slot machines?

Mathematicians have a formula for this, and it's called the Gambler's Ruin problem. Given an initial bankroll and the desire to increase it, what are the odds you'll succeed, assuming that you stop when you reach your goal or when you go broke ("get ruined")?

Here's a nice little web page that allows you to run an applet that simulates the Gambler's Ruin problem, along with a concise statement of how it all works. The applet assumes that you make the same bet every time, and that there's a maximum number of times you'll play.

For example, let's assume you start with $10 and have a goal of doubling that amount. You decide to play roulette, and you bet on black each time. The odds of winning a given bet is 18/38, or 0.4736. If you were to do this a hundred times, limiting yourself to a maximum of 200 plays, the application says you'll win about 25% of the time. 67% of the time you'll be ruined, and 8% of the time you'll still be playing after 200 turns. Something to consider when you plan that next trip to Vegas.

I can't use this to determine the likelihood of Bennett's claim that he "more or less broke even" - even if I knew the odds of a given slot machine, it has differing payouts (the Gambler's Ruin problem assumes even money odds - if you bet a dollar and win, you get a dollar back). Plus, this page only models the odds of winning. If you ask it to calculate the odds of finishing with the amount you started with, it assumes you never bother to play. Alas.

Here's a simpler application of the Gambler's Ruin problem, for two players. Here's an application of the math for dedicated casino players (see this article for a definition of the terms he uses).

Anyone want to bet that Bill Bennett never understood this concept?

Posted by Charles Kuffner on May 07, 2003 to Technology, science, and math | TrackBack

Given that Slots-Of-Fun Bennett was playing the $500 machines, the probable typical payout is 92- 95% on the slots and 93-97% payout on Video Poker (depending on the version you play). Note that most casinos use the tightest machines on nickel games (typically 80-85% payout) and loosest on dollar games. Video Poker is 3-5% higher payout, however, this will again depend on the version of the game (Jokers Wild, Jacks or Better, Deuces Wild, Bonus Poker, etc.) Las Vegas casinos usually post their payout percentages, while Atlantic City casinos are required to have a minimum 83% payout.

Actually, roulette is one of the worst games in the casino with a house edge of 5.25 percent on average (the 0 and 00 provide the edge), and is not the game of choice for so-called smart gamblers. These individuals go for poker (the table version), blackjack, and baccarat.

The best book I have read on the subject of gambling is Scarne's Complete Guide to Gambling (2nd edition, 1974). While the book is old, the mathematics that prove the house edge behind most casino games are worked out.

In the end, all gamblers die broke.

Posted by: William Hughes on May 7, 2003 2:02 PM

Brad DeLong talks about the statistics of this (here and here). He also links to another simulator showing that, ultimately, the house odds prevail.

A side note is the percentages being bandied about bear no resemblance to the 8-liners you see around town. Depending on how greedy the owner is, the payout may be as low as 60-65%. As an acquaintance (who owns more than a few machines) told a cop, "You can't call it gambling. You know you're going to lose."

Posted by: Charles M on May 7, 2003 3:28 PM

Apparently Bennett played often and played a lot over a long period of time (ten years). That gives you a large number of trials.

All you really need to know to have a high degree of confidence that he did not break even is the central fact of all machine play. Whatever goes in only a portion comes out.

BTW, since Bennett did it to "relax" why is $500 per pull more relaxing than say $25 per pull?

Posted by: dwight meredith on May 8, 2003 1:41 AM

Mil $ stats problem I'm vigorously trying to resolve - Is it worth chasing break-even?? Any ideas/suggestions would be most appreciated.

Hello Kuff / All

I think I have an interesting question to pose
to you if you mighthave a mere couple minutes to consider it.

In Las Vegas as you probably know wagering on
sports is legal. Basic wagering on baseball and
hockey is unlike football where there is a spread.

For these sports there is a "Money Line" and
for Favorites the convention is they are designated
by a minus (-) sign. -200 means you wager $20 to
profit $10. If you wager standard $20 wagers
you must win 66.67%(rounded) of the time to break
even over the long term.

What I do is parlay my wagers so if two wins
occur in a row I put $5 in a "Profit Pool" and I
have two more wagers of $20. Thus I have a tree
system - what I call Fav Tree.

IFF! (If and only if) the Favorite winning percentage
is a mere couple hundreths over standard wagering
break-even my profits and "yield" ($ profit /
$ wagered) ... "blows away" standard wagering profits
over the long term. IFF you can be a mere couple
hundreths over break-even I can readily demonstrate
by simulation your yield will be one beautiful thing.

The problem is picking consistently over the
long term even a smidgen over break-even for any
given money line -AND- Fav Tree won't work below
break-even ... because that is negative expectation
territory. No matter what gyrations I do in
"The Land Of Negative Expectation" my long term
expectation is a loss.

From my historical money line studies/observations
especially for baseball (MLB) it seems there's always
one or two money lines over the long term that have
a "good population" showing just a tad over standard
wagering Favorite Money line break-even for the entire
season ... thus positive expectation.

From my (corrected) Fav Tree simulator I see that
I need a mere few hundreths over "standard break-even"
in the -110 to -200 Money Line range to realize a
MUCH! higher profit and yield than standard wagering
after many iterations.

Higher Money Lines don't work because there is either
no or not enough money left over for the "Profit Pool"
on a split.

My idea is after the first week or so is to
"follow" which money lines show a tendency
for a winning percentage to remain the slightest
over break-even with a "good number" of
"occurrences" that make up the ratio.

I realize the danger that past performance is
no indication of future results and a given
money line may indeed start to "crap out" (no
pun intended) at any juncture. What was a
good Favorite winning percentage at a given
Money Line can "go south" at any time.

However that said, I also see that often the Money
Lines with the winning percentage over break even
with a decent number of "occurrences" can get to
points where one or two games don't change their
status of being over break-even.

My idea is to "follow the flow" that overall I'll
always be on the whole a mere few hundreths over
break-even ... that's all I need.

I was wondering how sound my logic is and is there
any mathematical formulae, principles and/or
criteria where it is valid/viable to do such
Favorite winning percentage "chasing"; perhaps
wagering a little at first and progressively/
proportionally (?) more as the winning percentages
for all the money lines becomes increasingly
established as the season wears on.

Thanks in advance for your consideration. I welcome
and look forward to your correspondence.


Joel Shapiro

P.S. FWIW If you're intersted in experiencing my
Money Line Fav Tree simulator and the phenomena I
hope I've sufficiently described in this document
firsthand you can download it from my web page at

[simulator link]

self extracting .zip file.

[Ctrl][l] (Lower case L) hotkey combination to
initiate macro.

Bolded blue cells are user defined.

You should be able to figure out the sheet and
what's going on from context.


Posted by: Joel Shapiro on April 5, 2005 8:24 PM