Frank Floor Talk: Book Review — Probability & Statistics

Frank Floor Talk: Book Review — Probability & Statistics

  • Buddy Frank, CDC Gaming Reports
August 23, 2022 12:00 PM
  • Buddy Frank, CDC Gaming Reports
  • United States
  • Other

The third review in this series about math books that will help you develop your casino skills will be a bit different. Instead of reviewing a specific book, we’ll review the subject: Probability and Statistics (we’ll call it P&S from now on).

For as long as I can remember, I’ve known that the odds of getting dealt a Royal Flush on the first hand of a standard video poker machine were 1 in 649,740. Sounds like quite a long shot, but anyone who’s worked a casino floor will verify that it happens all the time. Why? Because with millions of handle pulls a day, 649,740 rolls around quite often. But what’s the math?

 width=There are 20 cards that could make up one of the cards of a Royal Flush on the first card dealt from a 52-card deck (A, K, Q, J, 10 from any of four suits). After that, there are only four possible cards remaining for the RF out of just 51 (since the one suit was established by the first card, and now a card is gone from the deck). Next, there are just 3 out of 50, and so on.

If you multiply 52 x 51 x 50 x 49 x 48 you get 311,875,200. That’s how many different hands there could be dealt.   Then multiply 20 x 4 x 3 x 2 x 1, which gives you 480. That’s the number of ways of making a Royal Flush. Divide the first number by the second and you get the magical 649,740. The same type of math will let you know that in Draw Poker, where you get a second chance to make the Royal Flush, the odds change dramatically. If you didn’t get a single card of the Royal (and discarded everything) your odds increased to 1 in 1,383,484. But, if you were lucky enough to get three cards of a Royal Flush on the deal, your odds of hitting the rest of the RF are just 1 in 1,081.

Advanced P&S students know there are a few other ways to calculate these odds, but no matter the methodology, it should be obvious why understanding basic P&S and the math behind the permutations, variables, sets, distributions, Venn diagrams, multiplications and divisions are important to anyone in Gaming.

While many Arts & Science college students avoided majors which required mathematics, almost all graduates were forced to take a course generally labeled as “Introduction to P&S” or the generic “P&S 101”.  For many, this course put a blemish on their near perfect GPAs, but it also led many others to success in business or opened a new career path in marketing, artificial intelligence, and/or data science.

Simply put, “probability” is the study of how likely, or unlikely, something could occur. It is the bedrock upon which every casino operates. Rather than predicting outcomes, “statistics” is the science, interpretation, and analysis of those outcomes (or any other data) by looking at their frequency, patterns and magnitudes.   Those are my definitions, so they may differ a bit from those more qualified experts cited below.

I’ve taught “Basic Slot Machine Math” at both the University of Nevada, Reno and San Diego State University for over 25 years. If there was just one concept that summarized that work, it is this:


A simple example would be rolling six-sided die. If I offered you a $1 bet to predict the outcome (1-to-6) and promised to pay $5 if you were right, the formula would be:


Clearly, that’s not a great game for you, but one that could be an excellent supplement to my Social Security income. That’s about as simple as “probability” can be. Virtually every book I’ve ever read on P&S begins with a casino-based example; be it a dice game, cards or a roulette wheel.

Of course, the more variables you add, it can get quite a bit more complicated. A slot machine PAR sheet lists the “Pay tables And Reels” and charts the frequency of outcomes and payouts for all slot machine combinations. For an antique mechanical slot, the PAR sheets were seldom more than one page long. Today, a PAR sheet for a video slot can easily be dozens and dozens of pages with thousands and thousands of calculations. But they are all based on the simple formula above.

If your career goal is a mastery of slot operations, this should make you realize that you need to know P&S. “Statistics” will also be very useful when you’re trying to understand when and why “probability” seems to be wrong (is the customer lucky, the machine or promotion malfunctioning, or is the player cheating?).

But the biggest practical casino application of “statistics” these days comes from the marketing pros. Those who master the subject (or buy a good software solution) are better at consistently designing profitable promotions, productively modifying them and targeting them to the specific groups likely to produce the greatest returns. All of the leading casino analytical software programs are based on the fundamentals of P&S.

So how do you learn P&S if you’re new to the field? I was fortunate enough to take my first class from the late Bill Eadington at the University of Nevada. Before he became the country’s leading gaming economist, he taught P&S at UNR’s Business College. That course led both of us indirectly to long careers in Gaming. For most, taking a college course is probably still the most efficient way to learn the subject.   That’s assuming you don’t excel learning from a textbook alone (there are dozens and dozens of good ones on the topic).

However, today there are also some great online courses which bridge the gap between in-person classes and books. Surprisingly, many of these courses are offered in the $15 to $20 range and more than a few are free. Two good ones that come with a bit of snob appeal are from MIT and Harvard. While those courses can be audited for free, for an extra $139, the folks at Harvard will give you bragging rights and a prestigious (but expensive) “verified certificate” to hang on your wall.

So, with absolutely no math to back up the statement, but with a high degree of personal certainty, your odds of succeeding in casino operations are far greater the more you know about P&S. Hopefully, none of our current customers will ever master the topic because then they will realize that ……………. never mind.