ACKNOWLEDGEMENTS

FOREWORD by Anthony Curtis

1   A MATHEMATICIAN LOOKS AT CRAPS PROBABILITIES, EXPECTATIONS, AND BETTING SYSTEMS
     Chances of Winning at Craps
     Negative Expectation at the Crap Table
     A Gambling System

2   GRIFFIN ON BLACKJACK
     Casino Attitudes Toward Counters Changing He Thinks
     'Bonanza' on Early Surrender?

3   HOW LONG MUST WE WAIT FOR AN IMPROBABLE EVENT TO OCCUR?
     An Old Gambler's Rule of Thumb
     Some Examples
     Natural Log of Two
     Of Means and Medians
     Birthdays and the Next Issue

4   MEANS AND MEDIANS OF WAITING DISTRIBUTIONS
     How Many People to Get All Birthdays?
     But How Can We Find the Median?
     Next Time More Birthdays

5   MATCHING BIRTHDAZE
     At Least Two Matching Days
     Three or More People on the Same Day

6   A PARADOXICAL PROPOSITION
     A Build Up of Consecutive Sevens
     The Long Run
     Who Will Win Next?
     Other Strangeness

7   SHOULD YOU GAMBLE IF THEY GIVE BACK HALF YOUR LOSSES?

8   AN ALL-TIME HIGH--HOW OFTEN?
     Random Walk
     One Dimensional Random Wald as a Model for Gambling    Problems      
     Chance of Being at an All-Time High
     Next Time Games With Variable Payoffs

9   GAMBLER'S RUIN FORMULAS FOR NON-UNIT PAYOFFS
     Chance of Ruin Before Doubling Fortune
     Approximation Gambler's Ruin Formulas in Other Cases
     Games With More Than Two Possible Outcomes

10   OPTIMAL WAGERS ON SIMULTANEOUS BETS

11   TRANSCENDENTAL PROBABILITIES

12   DIFFERENT WAYS TO DESCRIBE WIN RATE (OR PC)
     Hold Percentage
     Throw Out Ties
     Disagreements on Blackjack Percentage
     Uston vs. ECON
     My Craps System With an Average Win Rate of 56%

13   DISCONTINUITY AT THE GOLDEN MEAN (A paradox of proportional betting)
      Golden Mean

14   SELF-STYLED EXPERTS TAKE A BATH IN RENO

15   MATHEMATICAL EXPECTATION FOR THE PUBLIC'S PLAY IN CASINO BLACKJACK
     I  Introduction
    II  Other Studies
   III  Basic Strategy, Definition and Implication for Casinos
    IV  How I Did It
     V  How Well Does the Public Play Blackjack?
    VI  Possible Objections
   VII  Per Capita and Per Dollar Win Rates
    Acknowledgements
    References

16   EXTRA STUFF
     Peter Griffin's Blackjack Quiz
        Answers
     A Typical Book Review
     Griffin Comments on 8-Spot Keno Ticket Story
     Guru Goof?
     So You Think You Know Craps by S.N. Ethier
         Craps Quiz
         Solutions to Problems
     April 1, 1983, Las Vegas

   INDEX

How Many People To Get All Birthdays?

     Now for something new. Recall the challenge at the end of last issue's article: How many people would you have to gather in a room to have at least an even chance that all 365 birthdays would be represented? What did you guess? I hope not 730 since, with 730 people, the odds are more than millions of millions to one against the event taking place. 
     You probably sense that what we're after here is a median, rather than a mean, waiting time. And yet our Old Gambler's Rule will be of no use to us because we're not dealing with a single improbable event, but rather with many events, most of which are quite probable. 
     Let me explain. After the first person enters the room we will then be waiting for another person to come in who has a separate birthday from the first one. If this were a problem of means we would now be on our way to a solution. The chance that a newly entering person has a different birthday from the first birthday is 364/365; hence we expect to wait for 365/364 people in order to get that second birthday covered. 
     Now imagine we've already covered two birthdays. How long (how many people) do we expect to have to wait until that third birthday arrives? On average it will be the reciprocal of 363/365, or 365/363. Once we have covered all but one birthday we will probably have to wait a long time to get the last one. How long? On the average it will be 365 people.

     Peter Griffin, renowned gambling mathematician and author of The Theory of Blackjack, has explored the mathematics of gambling for more than two decades. Extra Stuff is a collection of his most famous magazine articles and papers. 
     Extra Stuff addresses many of gambling's abstract concepts-proportional wagering, considerations for games with variable payoffs, the effects of rebates on losses-as well as more mainstream subjects, such as the casino's treatment of hold percentages and analyses of gambling systems. You'll also find book reviews, quizzes, letters to the editors of gambling periodicals-a virtual kitchen-sink-full of gambling information found nowhere else. 
     Extra Stuff-Gambling Ramblings is packed with mathematics, but Griffin’s easy style accommodates most readers. Gamblers of all levels of experience will come away with an enhanced understanding of casino games.