Author: Bob Dancer

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Two Cards or Four? When Does It Matter?  

Consider the following two hands in games where you get your money back for a pair of jacks or better:  A♠ K♠ Q♥ T♦ 5♣ and A♠ K♠ Q♥ T♦ 5♠. There are games where the correct play in both hands is AK; games where the correct play is always AKQT; and games where in the first hand you hold AK and in the second you hold AKQT. Today we’re going to look at which games fall into which category, and why.

First, I’ll provide the “executive summary” for when each condition holds. That’s all some of my readers wish to know. Afterwards, I’ll provide some more detail as to the “why.”

     A. In games where two pair returns 2-for-1, always hold AK.

     B. In games where two pair returns 1-for-1 and straights return 4-for-1, hold AK in the first hand and AKQT in the second.

     C. In games where two pair returns 1-for-1 and straights return 5-for-1, always hold AKQT.

 

Now let’s go a little deeper. In Category A, we basically have Jacks or Better and Bonus Poker. The pay schedule matters not at all for the value of AKQT, so long as it has the standard return for straights, two pair, and high pairs. The value of AKQT is always going to be $2.66 for the five-coin dollar player.

The value of AK, however, is also affected by the value you get for full houses and flushes — but not much. Holding AK in 9/6 Jacks or Better is worth $2.87 and $2.79 and in 8/5 Bonus Poker these values drop to $2.82 and $2.75 — which are still well above the $2.66 benchmark for holding AKQT. A key fact that will come in later in the article is that in the Jacks or Better case, when the low card is unsuited with the AK, then AK is worth 21¢ more than AKQT, and when the low card is suited with the AK, then AK is worth 13¢ more than AKQT.

You do get a full house holding AK one time in 900 and flushes one time in 99 when the fifth card is unsuited with the AK and one time in 136 when the fifth card IS of the same suit as the AK. These are not big numbers. The pay schedules matter, but unless you get into a Bonus Poker game that is so bad that two pair only returns 1-for-1, you always prefer AK.

How much you get for two pair doesn’t affect the value of AKQT because the only hands you can get when you hold those five cards are a straight or a high pair. From AK, however, you end up with two pair every 22.8 times. Since that means an extra $5 every 22.8 times when you get one unit more, changing the value of two pair from 2-for-1 to 1-for-1 is worth 22¢.

Close readers might remember that in 9/6 Jacks or Better when the low card was unsuited, AK was only worth 21¢ more than AKQT, so why wouldn’t changing the value of two pair change the play? The answer is that in the games where two pair receives 1-for-1, the value of 4-of-a-kinds (especially aces) is much higher than the $125 you get in Jacks or Better. Even though you only only have a 1-in-16,215 chance to get four aces holding AK and drawing three cards, the difference in the payout from $125 to $800 is worth about 4¢.

In 9/6 Double Double Bonus, for example, holding AK is better in the first hand by 4¢ and AKQT is better in the second hand by about 4.5¢. To me, these numbers are plenty big enough that I’m going to pay attention to the suit of the small card in these hands. Whether they are big enough for you to reach the same conclusion is a decision you’re going to have to make for yourself.

The third category where two pair returns 1-for-1 and straights return 5-for-1 is really just the best versions of Double Bonus Poker. Since the straights pay 25% more than they do in most otherwise similar video poker games, it should be no surprise that you hold AKQT on both hands.

There are a few games that will not fit these rules exactly, and variations on games like Ultimate X, but in general these rules apply widely.

I picked AK in this case. I could easily have chosen AQ or AJ with exactly the same results. All three combinations are equal.

The rules for KQ, KJ, and QJ are different than the ones given here. Perhaps I’ll discuss them in another blog someday.

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Ed Thorp’s Wife

Ed Thorp is one of the fathers of blackjack card counting. Recently he made a nice donation to the Blackjack Ball, which ensures the continuance of the ball even after it’s founder, Max Rubin, passes away. As a partial thank you, Max told Thorp that he could invite his entire family to this year’s Blackjack Ball if he wanted, totally complimentary.

This year, Thorp brought his wife, three children, one son-in-law, and two grandchildren. Slightly before the dinner started, I sat down at a random table. One of my goals at the ball was to find guests for the podcast. Before dinner I was writing up notes before I forgot who the people on my list were and what made them interesting.

It turned out that the table where I had sat down was the one that the Thorps were planning on reserving for themselves. As there are only eight chairs at the table, I told them I would move. They graciously said that wasn’t necessary and they’d just go find another chair, which they did. So, I ended up sitting between his granddaughter, who had just graduated from Dartmouth College, and his wife, Catherine.

While I conversed with both ladies, as well as others at the table, I had the longest conversation with Catherine. She had married Ed Thorp, who is now in his mid-80s and still quite sharp, about seven years ago. Since I married Bonnie five years ago, there were certain similarities.

Catherine said Ed chose her mathematically! He had some mathematical model that told him that she was the one! I found that fascinating. I’m happy with my choice of Bonnie, but certainly did not use any mathematical formula to pick her out. So, I decided to ask Ed to tell me more about his methodology.

He smiled and said it’s not original with him and Wikipedia addresses this as the “Secretary Problem.” My explanation below is going to be highly simplified, but you can look it up in Wikipedia should you like.

The Secretary Problem was originally derived in the context of determining how many potential secretaries you should interview before deciding on which one to hire. The problem requires that you know the number of interviewees in the pool, n, and that you are able to rank each secretary relative to the others.

The key number for the optimal number of secretaries (or candidates for marriage) turns out to be n / e, where e is a mathematical constant whose value is approximately 2.718.

That is, when Ed was looking for a wife, if he thought he would have five years to find one and would meet eight potential mates a year, that would be 40 candidates. Doing the math, we have 40 / 2.718 = 14.7. This means that with the first 14 women Ed meets, no matter how desirable they are, he does not propose. He merely ranks each one relative to the others.

Starting with the 15th woman, he should propose to the first one he meets who exceeds the others in ranking. Assuming she accepts, that’s the one he should marry. If none of the last 26 women outrank the best one in the first 14, he should simply marry the last one.

(Presumably in the real world, if the 40th one was totally inappropriate, he would continue until he found one that’s reasonably appropriate.)

So, I told Catherine that I might forget her name before next year, but I was probably going to remember that she was Ms. 15, because she couldn’t have been in the first 14! She and Ed were both fine with that!

This was a pleasant chat and “intellectual” discussion with two charming people, but I have my doubts that this was actually the method Ed Thorp used to select his wife. Why? Because it’s extremely difficult to rank people.

Let’s say he’s met Karen, Linda, and Mary and assume they are all “reasonably” acceptable. One will be prettier, one will be smarter, one will make better lasagna, one will be more responsible financially (a definite consideration because Thorp is, I believe, a billionaire or close to it), one will be more compatible on religious and political issues, one will be better liked by his family members, etc.

How can you possibly give a unique ranking for these three women — let alone 14?  It’s virtually impossible that one woman could be superior in every aspect to the others. So, you’re going to have to make some kind of formula that tells you which attribute is most important. And these are real live women, who each come with lots and lots of attributes, each of whom have good days and bad days. This is not a catalog where you can pick and choose.

So, this strikes me as an apocryphal story that Ed and Catherine like to tell but was not actually used in real life. However he actually chose her, he did well. She appears to be quite charming and they are clearly very fond of one another.

The Secretary Problem, however, is an interesting theoretical concept about when to stop your searching. I’m glad I was introduced to it.

Author’s note:  After I wrote the above, I sent it to Dr. Thorp to get his feedback before I published. It’s possible I misrepresented the problem, and I was, after all, suggesting that he wasn’t being entirely truthful in saying he used this technique to find Catherine. He is one of my heroes and I didn’t want to insult him.  He responded:

Hi Bob,

 

We enjoyed your company.  Yes, a math model like this is fun to think about but the real world is so much more complex in this instance that one shouldn’t, and we didn’t, follow it.  

Best,

Ed

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The Twenty-Third Annual Blackjack Ball

The Blackjack Ball is an annual institution where about 100 of the world’s best gamblers meet up, socialize, drink too much, and compete against each other. The 2019 version occurred a few weeks ago.

The first order of business after two hours of drinking and socializing was to vote for the newest member of the Blackjack Hall of Fame. Six worthy candidates were nominated, and each professional player got one vote — with Hall of Fame members votes counting triple.

This year’s winner was Rob Reitzen. This is the way Rob was listed on the ballot.

There are few, if any, players that have won more money in the history of Blackjack than Rob Reitzen. From simple card counting to shuffle tracking/sequencing to bottom steering to advanced computer play, Rob has beaten games in more innovative ways than most professional blackjack players even know to exist.

The founding partner of CORE, which went on to become the largest and most profitable player-banking operation in history, Rob was featured in an Esquire Magazine article in which thereporter followed him and watched him beat Caesars Palace in Las Vegas out of more than $500,000 while using a sequencing technique he dubbed “The Hammer” on a single weekend!

Rob’s acceptance speech was somewhat lengthy. Host Max Rubin quipped afterwards that Rob must have thought he had a chance to win so he wrote “War and Peace” to read just in case.

The highlight of the ball is the test of 21 Questions. The five top scores get invited into the skills contest. Of these five, one player is eliminated quickly.

There were three people there who were ineligible to play: James Grosjean, Richard Munchkin, and myself — albeit it for different reasons. James and Richard have each won three times and have been determined to be “too good.” The first-place winner gets the Grosjean trophy, and the second-place winner gets the Munchkin cup.

The winner also gets an engraved Nebuchadnezzar (15 liters) of Luc Belaire Champagne, donated by Hall of Fame member Don Johnson who is an ambassador for that company.

I got included on this list for an entirely different reason as my blackjack career was short and uneventful. Although I did get to the final table at the Blackjack Ball once, I blew out immediately and didn’t end up in the money.

My contribution is that I submit a LOT of test questions. I submitted about 30 questions this year and they actually used seven or eight of my questions, depending on how you count. Since host Max Rubin has been a bit under the weather, he doesn’t have the time and energy to create as many questions as he used to and he’s grateful to me for supplying some. And, since taking a test where I had submitted a third of the questions would hardly be fair to the others, I’m not eligible to play. That’s cool. I get a kick out of composing questions that stump some of the smartest gamblers in the world — while other such gamblers get the questions correct.

Before the competition, we have a Calcutta auction. Host Max Rubin is a very charming auctioneer. He groups each of the 100 or so contestants into about 20 categories and we bid on who is going to win. Max teases, cajoles, insults, and praises various people in the audience in order to get them to bid. About $20,000 is raised and this goes to whomever owns the players who end up in the top four positions. Each player is allowed to buy up to 50% of himself back after the Calcutta and before the test is given.

The test is difficult — and questions can be about anything and everything. Some of the more interesting ones were:

  1. You are betting $100 on the pass line and place maximum odds at the Bellagio where 3-4-5 odds are allowed. On winning bets, what is the typical payout for the combination of pass line bet plus the odds?   

 

Answer: $700 — $100 for the pass line and $600 for the odds

 

  1. Donald Trump is the 45th American president. Express the number forty-five in roman numerals.

 

Answer:  XLV — this is not trivial. Many will say VL, which is incorrect

 

  1. In a National Basketball Association game, with 0.2 seconds left, the ball is out-of-bounds in possession of the team trailing by 2 points. The inbounding player throws the ball directly at the basket — and the ball would indeed have gone in except one of the inbounder’s teammates gently touches the ball when it is directly above the basket, 12 inches from going in. The teammate guides the ball into the basket. What happens?

 

  1. The inbounding team scores three points and wins the game
  2. The inbounding team scores two points and sends the game into overtime
  3. The inbounding team is guilty of offensive goal-tending, which causes them to lose possession and the game
  4. 0.2 seconds is not enough time for any of this to happen, so time runs out before any points are scored or there’s a change in possession

 

(Answer b: offensive goal-tending only occurs when the shot originated within the field of play. Since this “shot” originated from out-of-bounds, offensive goal tending could not happen.

Although the play originally began beyond the 3-point-line, the fact that the ball was touched directly above the basket turns this into a 2-point play.

The clock doesn’t start until the ball is touched by a player within the field of play — which in this case happened when the ball was one foot above the basket headed downward. 0.2 seconds is plenty of time for the basket to be made before the clock runs out.)

Those questions were three of the ones submitted by me, so naturally I think they are some of the most interesting. Here is one submitted by James Grosjean.

  1.   Playing six deck blackjack where the dealer stands on soft 17, which of the following starting positions has the highest EV.
  1. You have a hard 20 against a dealer ace before he checks for blackjack.
  2. You have an 11 versus a dealer 5
  3. You have a pair of aces against a dealer 4.
  4. You have a ten and a nine against a dealer 8

(Answer:  the exact numbers were given at the Ball, but the correct answer is a.)

One of the professional players only got four out of 21 questions correct. He was invited forward to receive a small “World’s Worst Blackjack Player.” The teasing was good natured, but he definitely got razzed.

The player with the most correct, 14, was Andy Bloch, former member of the MIT blackjack team and also an accomplished poker professional. Two players got 13 correct: Gemlo and Big Player. Since they want to be identified by their pseudonyms, I won’t identify them any further.

Coming in at 12 correct were four players: Arnold Snyder, John Brahms, Tony S, and attorney Bob Nersesian. Since only five players advance to the final game, we had to eliminate two of these four players. The way this was done was each of these four people had to name a casino that was formerly in Clark County but is here no more. Giving an incorrect answer, or a duplicate, eliminated you. There were probably fifty casinos named (e.g. Stardust, Riviera, Dunes, Landmark, Las Vegas Club, etc., etc. and finally Arnold Snyder and John Brahms gave incorrect answers and were eliminated.

On to the skills competition, which is held at a blackjack table. There was a stack of cards in the discard tray. Each of the five contestants needed to estimate the number of cards there. Worst guesser was eliminated — which happened to be Big Player. We were then down to four players.

Next test was a card-cutting competition. The 6♣ was placed at the bottom of the deck. Each player got to cut the cards. Munchkin then burned a card and dealt any number of hands that player named (from 1 to 7), with the object being to place the 6♣ face down in the dealer’s hand, e.g. as the dealer’s hole card.

The players all chose to deal six hands. Unfortunately, Andy Bloch cut the card four spaces away and was eliminated.

The third test was very unusual and was designed by James Grosjean. JG cut approximately one-inch squares out of the center of all twelve picture cards. Players had 90 seconds to figure out the rank and suits of each of the squares. Positive points were awarded for being accurate. Negative scores were awarded for inaccuracy. The mean score was negative! Gemlo did the worst and was eliminated.

It was now down to Tony S and Bob Nersesian and the contest was counting down a double deck. Richard Munchkin, who was running the final contest, removed three cards from each of the two deck stacks. The players had to determine what those cards were, using any card counting system they wanted.

Unfortunately, Bob Nersesian has never counted cards, whereas Tony S was a highly skilled professional for a number of years. Bob’s only chance was a wild guess. As soon as the clock started, he should have slammed his cards down and predicted his three cards would have a count of zero in HiLo.  He would win if there were three neutral cards, or one high, one low, and one neutral. He had a reasonable chance for success, whereas if he actually tried to count the double deck, he had basically no chance. There are actually better counts to use if you’re just going to guess, but Bob wouldn’t know about this.

Bob chose the wrong way to go about it. He tried to count the cards and ended up almost a minute behind Tony. Tony was accurate in his count, of course, and it was over. Although it didn’t matter, we asked Bob what he thought the count was in his deck. He said +6, which is an impossible number with only three cards using HiLo. Bob is a hell of an attorney but has never counted cards.

Congratulations to all our winners. We hope to have Rob Reitzen, Tony S, and Max Rubin on our podcast soon celebrating the twenty-third Blackjack Ball.

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A Question from a Reader

Many of you know that, in addition to being posted on bobdancer.com, my columns are published on the gamblingwithanedge.com website. This site allows for readers to post comments. One recent comment by someone who used the name “Jerry,” read as follows:

 

This is off-topic but I wanted to get your opinion on VP for Winners. I have been playing a particular game at a particular casino for over two years now with excellent results on about two million in coin-in on two years of win-loss statements. I have a degree in math so I know that this is a significant sample space. The game is rated at 99.256% on VPW. My results have been 97.133% (without handpay & freeplay); 100.925% (with handpay); 102.297% (with handplay & freeplay). Since this game includes a variable multiplier, is it possible that it has been under-rated by VPW?

 

The only game with variable multipliers on VPW is Super Times Pay. The return on 9/6 Double Double Bonus is indeed 99.256%, so that is the game I assume Jerry is talking about.

Jerry makes several statements. Let’s look at them one by one. First, he’s using Win-Loss statements as an accurate reflection of his actual play. Over the years I’ve found less than 10% of such statements to closely track with my own daily figures. Assuming that one from an unnamed casino provides accurate information is not an assumption I’m willing to make.

Second, is $2 million a significant sample space? It probably would be if you were talking about quarter Ten Play. It probably wouldn’t be if you were talking dollars or higher.

Remember multipliers only come about in this game every 15 hands or so. And the ones that do come about are heavily weighted towards the “lower end,” meaning 2x and 3x, while the higher-end multipliers 8x and 10x are fairly rare. According to the Wizard of Odds website, these are the frequencies for each of the multipliers:

Super Times Pay — Actual Multiplier Probabilities

MUTLIPLIER PROBABILITY EXPECTED
2 17% 0.34
3 33% 0.99
4 16% 0.64
5 24% 1.2
8 6% 0.48
10 4% 0.4
Total 100% 4.05

Adding the top two multipliers together, they occur 10% of the time, meaning every 150 hands or so. The mini-jackpot in DDB, called aces with a kicker, occurs slightly less than once every 16,000 hands. Remembering that the big multipliers only occur every 150 hands and each hand costs $6 to play, this means “one cycle” of aces with a kicker in this game is 16,000 * 150 * $6 = $14,400,000. Calling one seventh of one cycle significant is a misuse of the term. It might be a significant amount of play to Jerry, but mathematically it is insignificant.

STP comes in Triple Play, Five Play, and Ten Play. In addition to aces with a kicker, dealt quads and/or royals are important as to whether or not they come with a multiplier. Being fortunate to get dealt deuces, for example, with or without a kicker, with an 8x multiplier in effect, is going to give you a much higher positive result than average numbers predict.

I don’t know what big hands Jerry received, but I strongly suspect they included big hands with big multipliers. This is called “positive variance,” meaning that in the time Jerry has played, he has been luckier than average. It happens. Congratulations!

Jerry also mentions free play given by this particular casino. This is definitely not part of the VPW calculation but is an important consideration in the overall return of the game.

The calculation of the value of STP is fairly straightforward for a computer program. The Wizard of Odds site lists the return on this game as 99.26%, which is consistent with VPW’s 99.256% given that they are displayed with a different number of significant digits. I trust the figures of VPW and suggest that you should too.

Going forward, Jerry, you should assume your results will be 99.256% (assuming you play perfectly — which is also far from a given for most players). The free play will be additional. Your actual results over a period as short as $2 million in coin-in will not be the same as that, but that’s the best guess going forward.

Do NOT assume that your 100.925% results will continue. It COULD equal that in the short term, but it’s unlikely and you won’t know for sure until you play the hands.

I assure you that there will be other games and/or casinos where your results will indicate negative variance. It’s just part of the game.

Your figures imply that free play at that casino is 1.3%. Possible, I suppose, but pretty rare. A lot of us would like to know where a casino pays that much on a game that returns 99.256%. Also realize that a 2.3% edge on $2 million of coin-in implies you are ahead $46,000 at this casino. Many casinos will restrict you, figuring you are “too good.”

At most casinos you will be on their radar. If their free play is actually 1.3%, even on a 99.256% game, the casino is giving away the store. Eventually they will wake up. If you are someone who has been hammering this game, you will be the first one eliminated.

Players differ on how to react to such a good play. Some play as hard as they can because they figure it’s going to go away pretty soon and they better milk it while they can. Others believe that if they take small nibbles out of the game, it will last longer and give them more profit in the long run. You’re going to have to make your own call on this.

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So What?

The Las Vegas Advisor publishes a “Question of the Day” every day. People send in questions and the LVA gets someone knowledgeable to answer each question. Often video poker questions are shuttled my way.

In a recent QOD, someone noted that they were quarter NSU Deuces players at Green Valley Ranch Casino, which is run by Station Casinos. The players saw on the screen “advantage player tracking system.”  These players usually only played on multiplier days.

If they are getting a half-percent in players’ club rewards, this is a game that yields the player less than $3 per hour if they play perfectly. Basically, nobody plays NSU Deuces perfectly. I certainly can’t and estimate that fewer than one-in-a-thousand players can. My comment at the end of the QOD was, “It’s a real shame that casinos are doing this to quarter players. You might have a $3/hour advantage over them on the big multiplier days if you play competently. You’d think a casino could fade that.”

I received an email from someone who works in casino management and wrote, “Casinos no longer look at a $3/hour-winner player as just that – as their analysis adds $6 an hour (from all players) for free drinks and general maintenance items (bathroom, garage if parking is free) etc.”

This is probably true, although the $6 figure may vary from place to place. Casinos and other businesses have “overhead” expenses, and if that is going to be allocated on a per-player-per-hour basis, it’s going to be some such number.

Personally, I’m surprised that free drinks are included in this calculation. I believe that giving away free drinks is a money-making proposition for casinos. The more people drink, on average, the more they lose. Yes, the drinks cost the casino money, but the increased revenue from players more than offsets this cost.

Going with the $6 figure for now, it doesn’t change my feelings on this. If the player costs the casino $9 per hour, is that really a significant threat to the casino’s bottom line? Few players play completely accurately. Players bring friends along and their friends play even worse. Players have leaks, where even if they play video poker reasonably competently, they also bet on their favorite football team, and in that game, they are definitely a square. Even solid players go on tilt periodically and do some crazy stuff that usually costs them a lot of money.

I’ve found $200/hour opportunities in casinos. Sometimes I get restricted after the casino figures out that they are giving away the farm. I understand that. (I don’t like it, but I understand it.)

But there needs to be a threshold where casinos tolerate a certain level of “winner.” I personally think that threshold should be higher than $10 per hour.

Possibly I would feel differently were I a casino manager. But worrying about $10 per hour players often leads to kicking out losing players who have just gotten lucky recently.

If any winners at all bothers a casino, they can certainly remove games that give players a chance. But if those games are there, players should be allowed to play them.

Enough of my rant. I’ll go back to a regular blog next week.

 

Author’s Note:  After this article was prepared, I received information that the IGT brand for player tracking systems is called “Advantage.” Thus, “Advantage Player Track System Activated” simply means that the casino is using the IGT software. It does not mean necessarily that the system believes the particular players are playing with an advantage.

Does this mean my rant was misplaced? Perhaps. Perhaps not. I still believe that casinos should allow players with small advantages to play, and that Station Casinos are among the worst culprits at not allowing this to happen. I will never be allowed to see the actual data to prove this one way or another, but looking at the data from the outside, this is the way it appears to this observer.

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Couldn’t There Be More Exceptions?

I was having dinner with a buddy, Al, who received a text from someone we both knew, Sandy. The text showed a picture of 8/5 Ace$ Bonus Poker along with a query whether I published a strategy for it?

Ace$ Bonus Poker is just like Bonus Poker, except the four aces each have a letter A, C, E, or S on them. If those four aces spell ACES in positions 1-4 or 2-5, then the hand pays 4,000 coins rather than 400.

I told Al to text back that the game paid 99.407%, which is about 0.23% superior to regular Bonus Poker. The game should be played exactly like Bonus Poker except for the hand aces full (e.g. AAAKK or AAA44, etc.). IF the three aces are in the appropriate positions to get the bonus (which usually isn’t the case, but can be), THEN you throw away the pair and go for the bonus, which you’re only going to hit 1/47 times (the same as drawing to 4-to-a-royal).

After a while, Sandy texted back. There are some REALLY close plays in Bonus Poker, such as A♠ 8♠ 4♥ 6♥ 7♥. In this hand holding the 467 is worth $2.4098 for the 5-coin dollar player while the A by itself was worth $2.4062. The difference is only 0.36¢, which is only about a third of a penny. Couldn’t the ace in appropriate position add enough to change the play?

I didn’t think so, but I said I’d go home and work it out and get back to Sandy. Since Al and Sandy are friends, I did this for free. I get all sorts of requests to figure out stuff for strangers that I turn down or say something like, I’ll do it for a fee of $xxx. Although I am a teacher and I do answer a lot of questions for strangers, I don’t see my role in life to be a research assistant for everybody who has a question. So, I pick my spots, so to speak. (And, of course, if the problem is interesting enough, I turn it into a column or two which pays me a bit.)

On the hand in question, holding one ace, you end up with four aces 44 times out of the 178,365 possible draws. That’s one time in 4,053.75. This calculation should be easy to perform for anybody with video poker software who can divide one number into another.

The next question is:  How often does one “sequential” ace turn into four sequential aces? For simplicity, let’s assume the sequential ace is in the first position (which means the ace of clubs, which has a big yellow “A” on it). From here, we’re going to need to get the ace of diamonds (the one with a “C” on it) in the second position, which will happen 1/47 times. If that happens, we need the ace of hearts (the “E) in the third position which occurs 1/46 times, and if that happens, we need the ace of spades (the “$”) in the fourth position, which occurs 1/45 times.

Since they all need to happen together, we multiply them. That is, one sequential ace will become four sequential aces 1/47 * 1/46 * 1/45 = 1/97,290. Looking at it a little differently, since we get aces every 4,053.75 from that position, one in 24 of them are sequential and 23/24 of them aren’t.

The previous two paragraphs required some fairly simple probability calculations. Simple probability calculations are necessary in figuring out all sorts of video poker and other gambling problems. To succeed, you either need to be able to figure this out yourself or have someone who does and is willing to help you either free or for a fee.

So, from this position, once in 97,290 times, the four aces will be worth $4,000. Since the video poker software already assumes that these aces will receive $400, the bonus is worth $3,600. Once you get here, dividing $3,600 by 97,290 gives you $0.037, which is almost four cents. This is ten times as much as the difference between the two plays. So yes, there are some exceptions that my simplification misses.

I was actually surprised to find this out.

Still, to get the optimal value out of this, you need to look at all the hands that could be affected. Usually they are an ace with a 3-card straight flush with no high cards and one gap, such as 457 or 78T with external straight interference suited with the ace.

There are a number of qualifications in that sentence. For players who don’t seriously study penalty cards, this is pretty complicated. Many players aren’t going to be able to remember it precisely. If they have it written down somewhere it will take them some time to find it, look it up, and determine if the current hand qualifies.

So, I’m personally not going to take the time and energy to memorize this exception, at least at the current time. I rarely play this game. I’ve played 8/5 Bonus for several hundred hours in the past, but not in the past few years. I will have to relearn 8/5 Bonus before I play this game.

It is possible that Ace$ Bonus will become my “go to” game in a particular casino somewhere down the line, and in that case, I’ll relook at this article and reassess whether I’m going to add this refinement to my game. But in the meantime, in the few times I play this game, I’m content to use regular 8/5 Bonus strategy with the aces-full exception.

If you wish to make the opposite conclusion for yourself, be my guest.

Note: After I finished this article, I passed it by a player who graduated from Cal Tech to verify that my conclusions were correct.  He responded:

I agree with your math, and want to add that there is another way of looking at it, via https://wizardofodds.com/games/video-poker/tables/aces-bonus-poker/

“To create a strategy for this game, you can use my video poker strategy maker, but use a 94 for four aces when there are no aces on the deal. With one ace, in a correct position, use 110. With two aces, in a correction position, use 200. With three aces, in a correction position, use 440. These are weighted averages of 80 and 800, according to the probability of getting ACE$. If you have aces that are out of position then use 80.”

Once more, the wizardofodds.com site provides useful information. The way to get the most value out of this site on this particular problem is to enter in the appropriate number for aces and look at the exceptions to the basic strategy. The ones that say the right play is the ace and not some other hold are the ones you want to look at.

I appreciate learning about the Wizard’s elegant solution to this problem and am glad to pass it on to you.

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A Lesson from Improv

I regularly participate in Thursday night workshops for an improvisation group in Las Vegas. Improv is a form of on-stage comedic acting where the scenes are made up on the spot in response to a suggestion from the audience. Although the group I go to is “clean burning,” meaning the language is suitable for all ages, it’s optional with each group whether to follow those rules or not.

Improv requires quick thinking and a sense of humor. On average, the people who come to these workshops tend to be both smart and a smart aleck.  These are my kind of people! I enjoy the group.

I first became fascinated with improv while seeing some “Second City” performances on cruise ships. Second City is a group that started almost 60 years ago and is probably the best-known improv troupe in the world. On cruise ships, most of their shows are family-friendly, but sometimes they’ll have a late-night “adults only” show where the language and gestures are raunchier.

In addition to the Thursday-night workshops, once a month we have a showcase where the better performers in the workshop get invited to perform for audiences who pay $10 to watch. I like the idea of being onstage and so am trying to get invited to perform. There are more wannabe performers than open positions and selecting the actors to perform is a subjective enterprise. Sometimes I get invited and sometimes I don’t. I expect to be invited for the February 16th showcase, but we’ll see.

There are a variety of games that are played during these workshops and showcases. I am best at the ones that involve rhyming. As a writer, I’m sort of a “word nerd” and rhyming comes easier to me than it does to others.

Plus, I study.

In addition to a few games where you have to compose songs on the spot, there are two different games require using words that rhyme with one-syllable first names, so I have prepared a list of words that rhyme with John, Stan, Kate, Pete, and about 30 other relatively common first names.

For a name like Jack the others in the group can “spur of the moment” come up with rhymes like back, black, crack, hack, rack, sack, stack, track, and whack. My list includes such words as amnesiac, Adirondack, Antonin Dvorak, Honoré de Balzac, cognac, almanac, Amtrak, insomniac, anorak and a bunch of others that are a bit outside of the mainstream. I have such lists for 35 names that I review periodically, “in case.” If one of these names is suggested by the audience, I tend to shine, which I am hoping increases my chances of being invited to perform in the monthly showcase. It’s an unpaid gig but being onstage is a type of high for certain of us.

Almost six months ago, on August 21, 2018, I wrote in this column about a buddy and his two kids. One of them, “Jack,” a twelve-year-old boy, also participates in the improv workshop along with his father. The vocabulary of a twelve-year-old is quite a bit less than that of most adults — and especially smart adults of the type who show up in the workshop.

To help compensate for this, I’ve sent Jack a few lists of rhymes and other material.  When Bonnie and I join their family for dinner once a week or so, we frequently play one of the improv rhyming games. As a result, Jack continues to improve and is now a better rhymer than some of the adults in the group who don’t practice. I have enjoyed watching his improvement over the past few months.

I’ve attempted to enlist Jack as an ally in my goal of being called to participate in the monthly showcases. I’ve provided him with a short list of rhyming names where I think I’m better than average. If I’m onstage and we’re going to be playing one of these games, I want him to call out one of these names from the audience that the group onstage will attempt to rhyme. Since I’ll be prepared, I’ll tend to do better than average.

Somehow, either I haven’t explained this well enough to Jack or he is unclear on what it means to be an ally. The last time the opportunity to call out a rhyming name came up when I was onstage, he came up with “Cole,” which is a name for which I was not prepared.

Spur of the moment, I came up with mole, foal, goal, and dole — but since I hadn’t studied this word, words like patrol, parole, rock-and-roll, español, shoal, skoal, and Superbowl, among many others, were not at the tip of my tongue. (They are now. After a name is used in the group, they tend to be re-used.) I had an opportunity to shine, with my helper in the right place at the right time, and he threw me a curve ball.

When I asked him why afterwards, he said he didn’t remember the words on the list I gave him plus he figured I was smart enough to figure it out on the fly. I’m not sure whether his “smart enough” comment was meant as a compliment or a dig. I’m betting on the latter.

While it is true that I could do better at that word than Jack could (assuming he had not extensively prepared for it), or better than many others in the group could do on the fly, I want to shine. I want to be a LOT better at these games than the others. I have only limited opportunities to impress the decision-makers with the fact that I deserve to be in the showcase. I want to be successful when these opportunities come.

So, I’m now asking Jack to review out loud the words I’ve sent him while he and his father drive in to the workshop every week. If Jack and his father, who’s also in the workshop, are both reminded of what’s on my list just before the workshop, perhaps one of them will “come through” for me. I’ve told them both that if they provide me with a list of the words they think are best for them, I’ll try to call one of them out at the appropriate time. I’m willing to be their ally whether or not they return the favor, but I strongly prefer that it be a two-way street. (Jack’s father is definitely an ally of mine — in many ways. Jack is so far a bit unclear on the concept. But he’ll learn!)

So, finally, what does this have to do with video poker? I am a very strong player and so some people conclude that I can play any video poker game in the world competently. Those people are incorrect – big time. There are a number of games I play at the professional level, but there are several dozens of games a casino could offer, along with a good promotion and slot club, at which I’m not currently competent. I can get good fairly quickly, of course, but it’s a process I must go through each time.

Right now, for example, I don’t play 9-5 Super Double Bonus competently. I used to know it well, but it’s not offered at casinos I frequent, for stakes I’m interested in playing, during the right promotions. So, the exact rules for the correct play when you have an ace along with a suited jack-ten are not front and center in my mind. I know sometimes you hold the ace by itself, sometimes you hold AJ, and sometimes you hold JT. Just knowing those things puts me ahead of most other players, but I want to be perfect at these decisions.

When I started playing Jacks or Better Multi Strike recently after a hiatus of a few years, I spent several hours getting up to speed and learning all the fine points. Now, I’m competent at this game, and so now is when I’ll go into the casino to play it during the right promotion. If I take another break from the game, I’ll need to relearn it one more time. Each time I relearn it takes a little bit less time than previously, as I certainly remember much of the strategy even after not playing for a while. But the fine points don’t stick with me as well as they did when I was younger.

The secret to my success, such as it is, is preparing extensively before I compete. This is true in video poker, improv, and many other things.  I like to “stack the deck,” as it were, so that I’m only competing when I’m prepared. Competing in things where I’m not prepared to succeed is not as attractive to me.

Obviously, none of us can be good at everything, and sometimes you must participate in things where your skills are less than average. I’m less than average at far more things than those things at which I’m better than average. But if I’m required to participate in something over and over again, I usually find it is worth the effort to get good at it. If I can’t get good at it, I’ll usually stop doing it if possible.

Author’s note: If you live in Las Vegas or are in the Vegas area on a Thursday night and are interested in checking out improv, look at //www.lvimprov.com. It will also provide you with information on our monthly showcases, where you are likely to find me, either onstage or in the audience.

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The Way You Walk

Assume the following:  You are a very strong player. In a particular, large casino, they only have two machines of the type you consider best. There are other players who play these same machines.

When you arrive on the property, how fast and purposefully do you walk to get to one of the good machines? Do you go straight towards the good machine or take some detours?

FAST:   There are good reasons to walk fast. After all, if one of your competitors for the machine is coming through another door to reach the machines, the early bird gets the worm. If the machines are full, you want to be the first one there to see if you can maybe make a deal to have a machine when one of them is finished.

NOT FAST:  There are good reasons to meander through the casino, taking your time and going in a roundabout way to your machines of choice. Most people do not rush through casinos (other than maybe to get to a bathroom). Rushing catches the eyes of observant employees and others. A slot director could well figure that if players are rushing to get to particular machines, perhaps those machines or the promotions should be looked at more closely.

When Math Boy visited GWAE not so long ago, he commented in passing that he can often tell whether a player is an AP or not simply by watching him move through a casino. We’ve had a number of guests on the show talking about tells at the poker table. This is a tell of a different sort, but just as easy for the knowledgeable person to read.

There is no such thing as knowing that nobody is watching when you’re in a casino. To be sure, much of the time you’re invisible. But not all the time. At most places, the cameras are always on, but often the output from a particular camera is ignored.

I make the assumption that everything I do in a casino can be seen. If I find abandoned credits on a machine, I turn on the change light and wait for an attendant to come and claim it. Although I’m in a differently category than many of my readers, I assume some people in every casino are looking for a good excuse to restrict me. So, I go out of my way not to give it to them.

At many casinos today, camera output is saved and they can go back several days later (or longer) and look at the tape should they so desire. Usually they don’t. But . . .

At the places with surveillance that’s the biggest threat to competent players, surveillance operators are trained to look for things that are “off.” Often such things are saved in the archives to see if it happens again — or has happened before.

Just walking purposefully is not a crime. In and of itself, it’s not particularly suspicious. But it’s unusual in a casino. If you show your act enough times, eventually counter measures will be taken against you.

Recreational players don’t particularly need to be concerned about this. You can do what you want. You will be giving off enough signs that you’re not a threat to a casino to be bullet-proof. Stronger players at least need to think about this. Are you giving the casino “extra” information about your play that they could use to restrict you?

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Worried About Blowing Your Cover?

In a recent Gambling with an Edge episode, Richard Munchkin and I were interviewing “Math Boy,” a Ph. D. in mathematics who for years used that knowledge to beat casinos. Math Boy told of the time in 2003 or 2004 when he met me despite his best intentions.

Math Boy wanted to stay “low profile” to other Advantage Players (or APs). The reason for this was that if he was playing a game, he didn’t want the rest of us to closely check out what he was playing and maybe decide that if Math Boy thought it was a good play, then maybe we should sit down too.

I understand this desire for anonymity, but since tens of thousands of people have attended one or more of my classes, any desire that I personally have for this kind of invisibility has long since vanished.

I have no recollection of this event, but I believe Math Boy’s version of the events. What he claims I said sounds very much like something I would say. I was playing Deuces Wild on the long-gone $1 Ten Play NSU machines at Harrah’s New Orleans. There was a lady sitting next to me, Math Boy’s wife, that I’d never seen before. Very likely there was no communication between us. I usually play quietly and concentrate on the game. Since I didn’t know the woman, there wouldn’t have been a friendship issue that could have sparked some conversation.

Math Boy recognized me and wanted to get his wife away from me without me taking any notice of him. He told his wife that he was hungry for lunch and wanted to go to the buffet. She told him that they would be serving breakfast for another half hour and then they could go to lunch.

Math Boy responded that if they went right then, they could get in for the breakfast price and after a few minutes they’d put out the lunch food which they could eat while paying the lesser price. Although it had nothing to do with me, Math Boy said I piped up and said: “That sounds like an advantage play to me!”

Math Boy went away cursing himself. He figured he had outed himself and that I then knew he was an AP. For the last 15 years or so he has believed that I have known who he is and have been on the lookout for him to find out what he’s up to.

He’s been worried about nothing.

Although I use the same timing ploy sometimes to get buffets for a lesser price, that is hardly evidence that somebody has all the skills to successfully exploit casinos. This is a fairly elementary move that many, many people know about. I probably didn’t even look to see who made the comment because it certainly wouldn’t have told me this penny pincher for lunch was knowledgeable gambling-wise. Or even if he were knowledgeable, he might well play for different stakes, or even play a different game.

If he played, for example, $5 15-9-4-4-3 Deuces Wild, which pays 0.8% less than NSU, I would have ignored him. At that time, I didn’t know that this “lesser” game had a MUCH higher theoretical, on some of the machines anyway, and you’d receive MUCH bigger mailers if you played it. There were possibly other players in the casino that day whose presence at that machine would have caused me to sit up and take notice. But not Math Boy, as he was “invisible” to me.

Sometime a few years later, another player I respected told me about the difference in theoretical. Possibly Math Boy knew this at the time, and possibly he played that game that day, but I need far more evidence than being frugal at the buffet to realize that he might know about the theoretical of the various games.

Apparently, though, Math Boy fretted about this event for many years. Too bad. Still, a lot of us regularly take precautions because we never know when somebody else is paying close attention. Better to take precautions when it doesn’t turn out to be necessary than to fail to take precautions when it actually does matter.

Author’s note: I sent a copy of this to Math Boy and asked if he wished to comment on it. His response was:

Sounds good. Only I live in a stochastic universe. I wouldn’t say I fretted about it for years. Or that I was sure I outed myself. I knew there was a probability that I might have outed myself. I’ve seen you a few times over the years and you’ve never looked at me with recognition. So the probability went from perhaps 20% down close to 0% over about a year.

Good. I’m glad he didn’t fret about it. Still, since this type of thing applies to a lot of situations and there just might be a lesson there for some players, I decided to leave it as written.

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A Story from Frank Kneeland

For the first six months of the Gambling with an Edge podcast, my co-host was Frank Kneeland. One of Frank’s claims to fame, video-poker-wise, was that he managed a large team of video poker players for two years in the early 1990s. On one of the shows, about six years ago, Frank told a story about a player on his team way back then.

Video poker teams at the time primarily chased progressives. If a game was breakeven at $8,000, his team might start playing at $11,000 or so (every team had its own strike numbers.) The team members would get a $10/hour “salary” and some percent of the royal if they hit it. The royal money would go to the team owner, who would pay all the taxes — minus the expenses of paying his team members. The tax laws have changed since then, and video poker teams are far less common than they were then.

I may have a few of the details incorrect in the story, but I’m sure I have the gist correct. I’m sharing it because it provides a lesson today

The player involved, “Joe,” had played on Frank’s team for several months. As far as Frank was concerned, Joe was the perfect employee. Joe would show up when called and stay as long as needed. Some players are “high maintenance.” Not Joe. He quietly played and rarely had any special requests.

One day Joe asked Frank if he was about to be fired.

“Fired?” Frank asked. “You’re one of my best employees! Why would I fire you?”

“Because I’ve never hit a royal flush, and obviously the success of your business depends on hitting royal flushes. Since I can’t seem to do that, you must think of me as an undesirable employee.”

“Hardly!” Frank replied. “I really don’t care who hits the royal. That’s all pretty random. But if I have enough competent players playing when the progressive is high enough, I’ll end up ahead. So just keep doing what you’re doing and don’t worry about it.”

Why do I bring this up today? After all, I hardly ever play progressives.

Because there are a lot of players really concerned about their score. If they are ahead, they figure they are pretty good. If they are behind, they figure the opposite. (Of course, players who don’t keep track of how much they are ahead or behind don’t need to worry about this. They have other problems we’ll address another time.)

The key to success, as Frank realized, is playing when you have the advantage. That edge can come from the game itself (as it did back then) or from the slot club, mailers, promotions, or sometimes other things. If you’re playing when you have the advantage, and not playing when you don’t, on average, good things will happen.

You just don’t know when those good things will happen. Every player who plays long enough will go through the type of dry spell Joe was experiencing. I have. Several times. And I’m confident that with my skills and game selection, if it happens to me it will happen to you too. It’s just part of the game. Not a fun part, to be sure. But a part that will surely show up from time to time.

Joe’s “problem,” if you want to call it that, is not that he was running bad. Since he only played when and where Frank told him to, the team as a whole was playing with an edge. Since he was getting $10 per hour, plus a bonus that he hadn’t collected yet but would when he finally hit a royal, Joe was personally playing with an edge.

Joe had one advantage over the rest of us. His salary guaranteed he couldn’t go broke while gambling. (He could certainly go broke if he overspent his income, but gambling losses would never cause him to go broke.) The rest of us don’t have that safety net. At the same time, I wouldn’t want to play for $10 an hour.

Joe was giving up expected value for the sake of a guaranteed salary. That fits well with some people. Especially people without the skill and bankroll to successfully play the games profitably. But also people who are risk adverse — who just can’t stand to lose.

I don’t like to lose. But it doesn’t bother me very much. I’m confident that things are going to turn out pretty well over all and today’s score doesn’t matter too much.