Gaming Gurus: Strategy Expert Article Archive - Page 408

Additional Information on the Field Bet

Hi Stickman, One thing I would add to your article on the Field Bet is that the bet is resolved on every roll. Compare a $5 Place 5 bet (HA=4.00%) to a $5 3x12 Field (HA=2.78%) bet over a perfectly distributed 36 rolls. Place 5 wins $7 four times and loses $5 six times for a net loss of $2. Field wins $15 once, $10 once, and $5 fourteen times; and loses $5 twenty times for a net loss of $5. Van Hi Van, You bring up a good point...read more

Jest for Fun

Sign seen written on men's room wall in a Las Vegas casino: "It's only a gambling problem if you're losing." * * * * * Math teacher to gambler's son, "Billy, tell the class how much is two and two." Billy: "Snake eyes." * * * * * The Flamingo Casino in Las Vegas was built by the notorious mobster Bugsy Siegel...read more

Deal or No Deal

I have always loved playing games. I grew up long before the video generation and most of the games I played as a child were board games or card games. It didn't really matter. I just loved competing against my family, friends or anyone else I could entice into playing. I guess my love of games is why I also enjoy watching certain games shows on television...read more

Please, no! It's not right!

In baseball there are many "unwritten codes" that are supposed to be followed by the ballplayers. For example, if your team is destroying the other team, your base runners are not to steal bases. Don't rub the area of your body where you were hit by a pitch; doing so shows you are not manly...read more

RNGs and bingo cards on Class II slots

The following letter is about Class II slot machines. Class II slots, unlike the Class III slots found in Las Vegas, Atlantic City, Tunica and elsewhere, do not determine the outcomes of their spins independently. They do not have random number generators (RNGs)...read more

Triple-zero roulette

Anna was a casual acquaintance in college, a friend of a friend of a friend who I used to run into at the odd party. I was more than a little surprised to receive a message from her via Facebook — and then I saw it was about gambling. "At a charity casino night, they had a roulette wheel with three zeroes," she wrote...read more

How do Edge and Chance of Winning a Bet Differ?

In casino gambling, edge indicates the fraction of every wager the joint should get to keep. Probability of winning suggests how frequently bets should pay off. Higher or lower edge means that the bosses keep more or less, not that a bet is less or more apt to win...read more

Big dice, little dice

Hey Frank: I love your work. I'm in the process of finishing your book Casino Craps: Shoot to Win! I have a question. I've been practicing since August with the dice rig I built using some dice I bought at a liquor store. My girlfriend recently bought me some legit casino dice, which are much larger than the store-bought dice...read more

Taking Advantage of an Advantage: Part 3 – Kelly Betting

            In last month’s article we discovered that when employing proportional betting, choosing the fraction f of our stake that we should risk when playing a positive game is tricky.  In particular we noted that if f is a large number (between 0 and 1), virtual ruin is almost certain.  Even lowering f to 0.3 was not sufficient to ensure that we would not experience ruin.  Is there any way to choose f that makes sense?             The development that follows is not as rigorous as that presented by Kelly [3] and other practitioners.  It is less technical and more simplistic but does, I believe, convey the idea accurately.             We noted that by trying to optimize our expected return over all possible paths (see last month’s article for the definition of path), we included many paths that would terminate with our being unable to continue betting.  In these instances the longer we play, the worse things get.  The approach we now take is suggested by some of the discussion at the end of last month’s article.  If the game is positive, we want to be able to play it for a long time.  In such a scenario the law of large numbers (commonly referred to in naive terms as the law of averages) says that for a large number of trials, the ratio of wins to the number of trials will be close to the win probability with a high likelihood.  In symbols, w/n will be close to p.  Similarly, l/n will be close to q.             Another way of saying this is to say that np will be close to w and nq will be close to l.  How close?  You’ll have to get out your old probability book for that one  I am going to skip that issue though it is a real one (see [1],  [4], and [6]).  Recall that last month we derived the expression                                                 Sn = (1 + f)w(1 – f)lS0        (1) For large n we then have                                                 Sn ~ (1 + f)np(1 – f)nqS0     (2) where ~ stand for approximately equal.  We can easily rewrite (2) as                                                 Sn ~ [(1 + f)p(1 – f)q]nS0     (3) Defining the function G by                                                 G(x) = (1 + x)p(1 – x)q, 0 ≤ x ≤ 1     (4) we can rewrite (3) as                                                 Sn ~ [G(f)]nS0                    (5)             The approximation indicated in expression (5) makes it clear that the righthand side of this approximation determines how Sn propagates.  In particular, if G(f) is a number smaller than 1, then the right hand side of (5) will get smaller and smaller as n increases.  Similarly, if G(f) is larger than 1, then the righthand side of (5) will increase as n increases.  Because this expression approximates Sn we can draw a similar conclusion regarding Sn and the choice of f.             Notice that from expression (4) we see that G(0) = 1 and G(1) = 0.  What about values of f between 0 and 1.  Here is where a bit of calculus is handy.  The derivative of G, written G’(x), is given by the expression                                     G’(x) = [(p – q – x)/(1 – x2)]G(x)      (6) For those of you who have had calculus, I leave the derivation of (6) to you as an exercise; the rest of you will just have to take my word for it.  It is the interpretation of this formula that is important here.  If one were to draw a graph of G on the interval from 0 to 1, the formula given in (6) would give you the slope of the tangent line to the resulting curve.  Notice that G’(0) = p – q, which we have been assuming is positive.  That means that the function G(x) is increasing as x increases from 0.  We know that G(1) = 0, so G must reach a maximum at some point to the right of 0.  At such a point the tangent line to the curve would be a horizontal line and thus have a slope of 0.  Also G(x) would be positive at such a point; in fact it would be greater than 1.  Hence, since x is less than 1, 1 – x2 would also be positive.  The only way to make the slope 0 at such a point would be to take p – q – x = 0 or, in other words, to set x = p – q = e.             Here then is the Kelly Criterion.  Simply set f = e.  In words, the fraction of our stake that we should risk is equal to the advantage that we have.  In our running example this would be 2%.  Now we see why we had such difficulty last month.  The values of f that we chose were just too large.  If we set f = e, then G(e) is greater than 1 and is the maximum value that we can take for G.             A few words are in order here...read more

Who Needs Training?

There is no such thing as a bad player only an untrained one. That is my company motto. I thought up that motto based on long term observations while in the casino. In addition, if you go on the Internet or to the library as much as I do, you get to read a lot of data about gaming...read more