Understand the basics of gambling mathematics

| Friday, Nov. 11, 2011

When I graduated from high school decades ago, I wanted to be a math teacher.

Getting assigned to a college calculus class that started at 7 a.m. three days a week quickly ended that dream. It wasn't just that the course was intimidating; in my new-found collegiate freedom, I reasoned that no one should be forced to start thinking that early in the morning -- or even get out of bed.

Besides, who really believed the math teacher's mantra that this is the one subject you'll use every day of your life?

It turns out that Mr. Schneider, the high school teacher in tiny Seneca, Kan., who I wanted to emulate, was right. Math touches us every day -- especially those who gamble.

Taking the time to understand a little math can help you enjoy casino games more and make your money last longer -- or maybe even let you leave with more than you started.

Let's look at a few math concepts and how they apply in a casino. Don't worry about running into complex formulas; after all, I barely passed that calculus class.


The simple definition of probability is that it describes your chances of winning. It's typically expressed as a fraction or a percentage.

Consider roulette. The ball can fall into one of 38 slots on the wheel. The slots numbered one through 36 are evenly divided into black and red, while the zero and double-zero are green.

If you bet red or black, your probability of winning is 18/38 or 47.37 percent, while the casino's chances of winning are 20/38 or 52.63 percent. You'll get paid even money when you win, but you'll win less than half the time over the long run.

Each casino game has a built-in house advantage. That's why it's so difficult for a player to win over the long run. No betting system can overcome the house advantage.

In roulette, the house edge is 5.26 percent, meaning you'll lose an average of $5.26 for every $100 you bet. For the pass line bettor in craps, it's 1.41 percent; for Three Card Poker, 3.37 percent; for a strong basic strategy blackjack player, 0.36 percent; for slot players, about 10 percent.

If you concentrate on games and bets that have a low house advantage, your money will last longer.


Volatility refers to the ebb and flow of winning and losing; you can be ahead by a lot and then find yourself down.

Volatility is a factor in slot machines and table games alike. It's a big part of gambling's appeal.

Gambling mathematician Eliot Jacobson, who analyzes casino games, says the technical definition of volatility is the average distance of players' results from the overall house edge. If the range stays close to the house edge, a game has low volatility. If it varies from going broke to winning a huge jackpot, the game has high volatility.

A better way to understand the concept is to think of the ocean.

"You can have small waves, medium waves, big waves," Jacobson says. "Volatility refers essentially to how choppy the water is."

He says volatility is what gamblers "buy" in a casino.

"A casino game is simply an opportunity to put money into some black box and have something happen -- either have nothing come out, the same amount come out or a greater amount come out. That's what a casino game is in the abstract. The fee the casino charges for putting that money into the black box is the house edge.

"What a player is purchasing in that black box is the volatility of the game, that spectrum of wins or losses, that choppy water if you will."

Independent events

If the occurrence or non-occurrence of one event does not affect the probability of another event happening, the two are said to be independent.

Let's put it another way: The dice have no memory.

Whatever number a craps shooter just rolled has no effect on what the next number will be. The dice will land on one of the 36 possible combinations. The odds of rolling a seven are one in six and the odds of rolling a 12 are one in 36, no matter what number the shooter rolled the last time or the last dozen times.

It's the same with a roulette table or slot machine. Each spin of the slot machine has an equal chance of hitting a jackpot, regardless of whether the machine just hit or it hasn't hit for 1,000 spins.

If you flip a coin and heads has shown up five times in a row, the chances of the next flip being heads are still one in two. At this point, you're not betting whether heads will show up six times in a row; you're betting whether this particular flip will be heads, and what's happened before has no effect on the outcome.

German, 22, wins WSOP

Pius Heinz of Cologne, Germany, won the World Series of Poker's Main Event this week after more than six hours of heads-up play against Martin Staszko of the Czech Republic.

Heinz, 22, won $8.7 million, while Staszko got $5.4 million. The $10,000-per-seat Main Event started with 6,865 players.

ESPN broadcast the final table on a brief time-delay; the edited rebroadcast starts at 2 p.m. Sunday on ESPN2.

Money trail

Slot players lost $47 million at Pennsylvania's 10 casinos during the week ending Nov. 6, the Gaming Control Board said. That's up from $44.4 million in the comparable week last year.

The state takes 55 percent of the gross slot revenue -- what's left after all jackpots are paid -- in taxes. Statewide, slot machines have a 90.1 percent rate since the fiscal year started in July. For every $100 bet, the machine returned an average of $90.10.

Payout rates for Western Pennsylvania casinos:

  • 89.84 percent: Rivers; gross revenue for the week was $5.27 million, up from $4.81 million last year.
  • 89.73 percent: The Meadows; gross revenue for the week was $4.34 million, down from $4.59 million last year.
  • 90.46 percent: Presque Isle in Erie; gross revenue for the week was $3.19 million, down from $3.22 million last year.


I've read about "stepper" slots. What are they?

Stepper slots are machines with reels that actually spin, as opposed to video slots, whose displays are electronic. Some video slots have displays that look virtually identical to reel slots. Regardless of type, a random number generator determines which symbols appear.

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