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Hello Mr. Wizard. I've been visiting your site for almost a year and want to take this opportunity to thank you. Thank you not only creating the site, but for constantly updating it with new and informative material. I can only imagine the amount of time your must put into it.

I have a number of questions. The first is about the banners. I'd like to know how we could optimize your income by clicking on the banners. Do you get paid on the basis of hits, unique hits or a flat fee or some other formula? Currently I only click on a banner if it's something new, but I'd be happy to click on a few banners every time I visit, if that would help generate income.

The next questions I have are about your perfect strategy for Jacks or Better table, and the practice Jacks or Better game. I hate to seem dense, but that's never stopped me from asking questions before. Where on the table do you find the rank of a hand with 1 high card (A,K,Q,J) and no penalty cards? My second question is somewhat related to the first. According to the practice game, the optimum play for a hand with unsuited A,Q,K and no penalty cards is to hold the K,Q and discard the A with the low cards. Intuitively, I would have thought that keeping the Ace is the better play. What is the advantage of dropping the Ace, and how would I determine the optimum play on this hand from the table?

Denis from Rochester, New York

Thanks Denis for your kind words and your interest to help keep the site financially healthy. When you first wrote just clicking my banners helped. However, as I update my answer in 2013, it is acquisitions that put rice on my table. It doesn't do much good to click on the banner if you don't at least sign up for an account, and preferably make a deposit.

You can see from my 9-6 jacks or better strategy, that there is a single line for "one high card." That is because an individual jack to ace is worth about the same.

Please tell me where I can find a copy that I can print and carry in my purse on the odds for blackjack. I am just learning and I am not sure when to take a hit and when to stay.

Sue from Marquette, USA

One idea is to right click on one of my strategy charts, save the image, and then print it. Some casino gift shops sell basic strategy charts the size of a business card. However, these charts have at least one mistake about half the time.

Very interesting site, do you produce your income as a gambler? Have you looked at the probabilities in stock index futures trading?

Randy from Aurora, Illinois

I am tempted to learn more about futures trading but for now I simply don't have the time. My income comes from both advertising revenue on this site (please click through the banners) and consulting fees on the analysis of new casino games.

Great site! Is there any way I can download your pai gow practice program to preclude always having to play online? Thanks in advance.

Abraham from Ocean, New Jersey

Sorry, there isn't.

Are the craps probability numbers with the odds taken 100% reliable. Also is the gaming industry your full time profession, and do you visit Atlantic City often? Also, how do you simulate billions and billions of hands, spins, and rolls. Is it computer generated and if so with which software?

DB from New York, USA

Well, anyone can make a mistake, but craps is an easy game to analyze mathematically so I would be very confident my odds on craps are right. Yes, gambling in one way or another is my full time self-employed profession. I have been to Atlantic City many times in the last few years but two months ago I moved to Las Vegas. So, I'm afraid I wouldn't be gracing Atlantic City with my presence much any longer. I prefer a combinatorial approach as opposed to random simulations whenever I can. Either way, I roll my own software with Visual C++. For random numbers I use a Mersenne Twister.

The Stratosphere advertises poker machines that pay over 100%. In an earlier column, you said that in full pay Jacks or Better the perfect strategy player will average one royal flush every 40,388 plays. Given this fact, does this mean a player needs to play this many hands perfectly before the advertised payout percentage is realized? I speak for the millions of video poker players who, like myself, watch a $20 become $0 in that "98%" machine.

Derek G from Vegas, baby!

No, it does not mean that. Contrary to popular myth, there is no cycle. Every hand is independent. It would take an infinite number of hands, played perfectly, to guarantee reaching the theoretical 99.54% return.

Here are some figures for you. Royals contribute 1.98% to the return in 9-6 jacks or better. That means you can expect the game to return 97.56% between royals. The standard deviation of one hand is 4.42. The standard deviation of the return of 40,391 hands, the average number between royals, is 2.20%. So, even after a complete royal cycle you can still be a long ways from a 99.54% return. Thee is a 95% chance you'll be somewhere in the range of 95.24% and 103.85%.

About the pop-ups, I hate them too. However, something has to put rice on the table. Consider them the cost of the information you're getting.

Hi. Do you sell or recommend a basic blackjack strategy card. In the past I purchased some from Winning Publishing NV. Your Site is the very good.

Tom from Lancaster, USA

Your timing with this question is perfect. Marsha Ness, the proprietor of Custom Strategy Charts is now ready to sell laminated blackjack and Spanish 21 strategy cards. The strategies come directly from me and there is a wide variety to choose from depending on the particular rules. As of this writing, electronic payment options are not in place yet so you'll have to send a check by mail.

Update: Since the publication of this question, Custom Strategy Cards has gone the way of the dodo bird.

I thought you blacklisted the Golden Palace Casino but I just received an email from you promoting them. What gives?

Grey from Singapore

You’re not the only one to complain. My partner in charge of advertising thought my feud with them was ancient history and accepted the ad. I was not happy about it and I apologize for it. Hopefully this won’t happen again.

Do you sell t-shirts?

Dennis

No. I have thought about it but I’m afraid that if I ordered a large quantity I would get stuck with most of them.

Recently found your site, really nice. Online casino reviews has been removed, will it return?

Floyd from Mattoon, Illinois

I’m not sure. The site continues to be very disappointing in terms of advertising revenue. I speculate the reason is partially that viewers were ignoring the ads and going right to the free links in the reviews. In other words I was giving away too much free traffic.

Where can I buy nicely printed and lamenated credit card sized versions of your basic strategy tables for 21? If you don’t offer them--please do!

Steve from Tuscon, U.S.A.

I’m glad you asked. You can get them through Custom Strategy Cards. These are my own blackjack and video poker strategies on nice wallet sized laminated cards.

Nov. 13, 2007 update: The Custom Strategy Cards site is now defunct.

Your internet newspaper column has not been updated since June. Are you OK? I hope so. I have a Black Jack basic strategy card and it is very beneficial. Are there basic strategy cards for electronic poker games? Thanks for your time.

Charlie from Shaumburg, Illinois

I'm not sure what Internet newspaper column you are referring to. However I'm okay. As a matter of fact you can get my own video poker strategies in a handy dandy strategy card at Custom Strategy Cards. The video poker cards may not be listed yet but I know the proprietor has them so just ask.

Update: The Custom Strategy Cards business no longer exists.

Good job and well done. The question: I notice from your May 5, 2003 Column that you actually CALCULATE your blackjack odds. I am a bit surprised that you were not using your computer to SIMULATE the results. Or is this a stupid question, i.e., the computer will take a million years to do the job?

Anonymous

Yes, I calculate blackjack odds using a combinatorial approach, analyzing every possible ways the player and dealer cards can come out, taking the greatest expected value at every decision point. This is harder to program than a simulation but I feel is more elegant and a nice challenge in recursive programming. However I still respect my peers to do simulations. With today’s computers it doesn’t take long to run off a billion bets, which gets very close to the optimal strategy return.

I enjoy your site a great deal. This morning I was reading your reviews of online casinos and your experiences with them and it seemed you inevitably lost everything on your deposits. I found this discouraging. Then I saw that you requested donations and now understand why you do so!

Anonymous

Since I started gambling on the Internet four years ago I am up about $20,000, about half of which is thanks to the Golden Palace. What you’re seeing in my reviews is just my most recent experiences, and like everyone else sometimes I lose for a month or two. However I just won $786 at Casino.Net and $2317 at Casino Kingdom. So I don’t ask for donations to subsidize a losing gambling habit. Donations simply help me to keep offering this site to the public for free. Although in fact the donations come in through PayPal which I then usually use to buy things on eBay.

I tried to find your reviews of the online casinos but the page says you have stopped making them available. I personally like knowing if a casino has a questionable policy. Why did you delete the reviews?

Anonymous

That is a fair question. For those who don’t remember I used to review all the Internet casinos I played and had a list of at least 50 reviews. One problem in maintaining this list was I got numerous requests from casinos not on the list to be added. I received no revenue through these reviews and in fact they may have taken customers from the paying advertisers. Another problem is Internet casinos often rise and fall in quality. I was taking a beating on the bulletin boards for saying good things about casinos that had evidently fallen downhill. So I started to remove reviews there were over a year old. However I still found it a hassle dealing with the requests to be added and dealing with complaints coming from players who had a bad experience with a casino I gave a good review to. My site does not make that much money so I decided revenue did not warrant the bother of maintaining that section.

I was playing jacks or better on your site. I had the following: 2(spades),5(clubs),J(clubs),10(diamonds),7(clubs). The optimal play is keeping the Jack only and your program stated the expected value was 2.3662715, but I calculated it myself and got 2.3662714. Which one of us has the rounding error?

Anonymous

Leave it to a couple math geeks to argue over the eighth decimal place. I checked the exact combinations of each hand and you are right. The exact value is 5*(84412/178365), or 2.36627140974967 to 15 decimal places. Whatever you are using obviously carries floating point arithmetic to more decimal places than the internal Java calculator.

Congratulations on your upcoming 100th "Ask the Wizard" column. I anxiously await each new addition as it adds to my gambling knowledge. My official question is... Can I be in the 100th issue?

R.S. from Dallas

Consider it done! Yes, this is my 100th column. Four years and two months ago I started it as a way to freshen up the site and to show the content is not stagnant. As evidenced that it has gone on this long I think it was one of my better ideas.

How did you decide to simulate 5,197,920,000 games of 10-hands each for your Texas Hold’em page? Is that number significant?

Anonymous

I set my simulator to run a certain amount of time. Every 10,000 hands the program checks the time and when it has passed the ending time it stops wherever it is.

Mr. Wizard, Great site. There is a lot of useful and interesting info. I’d like to see more of the mathematics and possible sources of simulations (source code, books, etc.) behind the games. Where would you suggest that a person interested in writing something similar to your "blackjack house edge calculator" go for more info? Thank you for your response.

Anonymous

Thanks for the compliment. I’m afraid I know of no source, including myself, that shows code for game analysis. It took me years to get my blackjack engine to work perfectly (splits when the dealer had a 10 or ace showing was very tricky). An easier way to get the house edge for blackjack is to write a random simulation. One of these days I would like to write a book on how I analyzed the games, but I’m afraid only you would buy it.

Love your site, thanks for all the work you put into it. I try to click some ads every now and then though I don’t gamble online.

Anonymous

(I'm going to let my ad-man, Michael Bluejay answer this one.)

Thanks for trying to help us out but save your clicking, because it just wastes your time and doesn't help us any. Our advertisers pay us a flat rate per month so we get the same amount of money whether you click or not. But even if we did get paid on a per-click basis, we still wouldn't ask you to click on ads gratuitously, because that wouldn't be fair to the advertisers. Advertisers who pay for clicks are expecting to get business from those clicks, and it hurts them when people click with no intention of buying. Wherever you are on the Internet, if you know the advertiser is paying by the click, then it’s kind of mean to click their ad if you know you’re not really interested in checking out what they have to offer.

We’re unusual in that we charge advertisers by the month. Most ads for online casinos are affiliate programs, where the webmaster gets a percentage (35% or so) of what the players lose, after they click over and open an account. It’s actually of questionable legality for U.S. webmasters to run ads as an affiliate, which is one reason we don't. Another reason is that our players tend to be a little more educated and less likely to lose, which would cut into our affiliate commissions. So the big reason we don't do affiliate programs is that we don't have to -- as one of the premier gambling sites on the net, we're able to sell ads on our own terms because so many online casinos fall all over themselves trying to pay us for some of our limited adspace. It's good to be on top. :)

You have inspired me to analyze some games myself. However I don’t know how to program a computer. What language would you recommend I learn?

Anonymous

C++. To know C is to love C. Java, Perl, and Visual Basic are similar to C but in my opinion are not as elegant or powerful.

I have been researching Casino Bar because they have a nice bonus today. I ran across your claim that their software does the equivalent of "dealing seconds", but I see that your information was last updated about two years ago. I was wondering if you know of any change in that situation, please. I suspect you would have updated the page if there had been a change, but I thought I would ask. Where could I find a basic strategy sheet for a casino dealing seconds? (the bonus may or may not still be playable). Am I right in thinking the house edge is close to 5% in such a game? Of course, if that is the case I may as well play tricard poker. Thank you for a great website. Do you accept donations?

Ricardo from Malden, MA

When I find a casino is not playing fair I don't generally go back to check if they've stopped. Sometimes I do if requested by the casino and I feel the problem may have been accidental. Following is a basic strategy, based on infinite decks, where the dealer stands on soft 17 and deals seconds. What I mean by dealing seconds is that if the third card, and only the third card, would break the dealer it is skipped and the next card is played, whatever it is. Otherwise play continues normally. The house edge under this game would be 9.3%. I used to ask for donations but got so few I quit asking. Now the site is comfortably supported by advertising revenue anyway.

Hi, fantastic site, keep up the good work :) I’m trying to work out risk of ruin statistics for any given combination of number of betting units and number of hands to play. Currently if the pair of values don’t appear in your RoR [Risk of Ruin] table in the blackjack appendix then I have no accurate percentage chance of ruin. Would you publish the formula for us please? Many thanks.

Hector from Cardiff, UK

Thanks. I explain how I calculate the risk of ruin in video poker in my video poker appendix 1. However for games where the bet amount is not always the same the calculations get very messy and computer simulations are necessary.

In two different sections on your site, you mention that in general it is better to bet on underdogs rather than favorites against the money line. However, as you point out in your Sports Betting Appendix 3, the money line house edge is LOWER when betting on the favorite. Isn’t that a contradiction? Isn’t it true for a bet of any type to seek out the lowest house edge?

Edward from Placentia

Good point. After giving it some thought I have decided to remove my sports betting appendix 3. It was based on the assumption that the fair line is exactly between the two money lines. For example in the last Super Bowl the money lines were generally Seattle +160 and Pittsburgh -180. My appendix was based on the unrealistic assumption that the fair money lines would be +/- 170. In reality money lines are market driven. Square action on the money line tends to favor the favorite, creating value on the underdog. Assuming Pittsburgh as a 4-point favorite was fair historical data suggests a 4-point favorite has about a 61% chance of winning. That would make the fair money lines +156 on Seattle, and -156 on Pittsburgh. To remind you most casinos had the money lines at +160/-180. Of course Pittsburgh won the game this time, but historically speaking I’m sure you would make out better betting underdogs on the money line over favorites.

I am continued to be amazed at the number of personal relationship questions your readers ask. Your answers are spot-on, as far as I am concerned, but does it surprise you that you have become a modern-day "Dear Abby"? Maybe there is a correlation between solving complicated math problems and solving complicated matters of the heart?? In any case, keep up the good work.

Scott from Pittsburgh

Thanks for your vote of confidence. According to my count I only received four such questions in the first 134 columns. However starting in August 2005 they flooded in, partly because my webmaster, Michael Bluejay got us the #1 rank in Google for a search on "Is my boyfriend cheating on me". We have since dropped to #2. To answer your question, yes, there is a connection. I take a cold calculating look at both casino games and life. When I give advice in either venue it is based on what I think will make the asker happier in the long run.

Are you printing those questions from those dumb bimbos who write in (some of them can actually spell in English) about their troubles with their boy friends strictly for comic relief? Enough already, I would much rather see you devote that space to real questions about casino gambling.

Jack from Troy

Your opinion is duly noted. The advice questions take me very little time to answer, while the gambling questions can sometimes take hours. I answer every worthy gambling question I get, so the advice questions are not taking from the gambling questions. However you are not the first to complain so I will segregate the questions starting now, so you will know when to stop reading.

In your last column you said that "The probability of 5500 coins not being a 55 can be very closely approximated as 0.9864012865500 = 1 in 507,033,772,284,213,000,000,000,000,000,000."

I assume "approximated" because of the effect of removal as you go through the 5500 coins. Talk about a minuscule effect of removal! This is a good example of the target coins becoming LESS likely as you remove non-targets, because the effect of removal is so small compared to the much larger probability of a crooked game, i.e. the target coins have been removed.

Pete from NY

Yes, I said “closely approximated” because there are only so many pennies in the world. Remove one non-55 from the bag and the effect of removal increases the probability that every other penny in the bag is a 55. If I hadn’t said “closely approximated” at least three people would have written in to correct me. It is of course an extremely minute effect, but many of my readers are perfectionists, and will jump all over me for the slightest of errors.

Great site! Your blackjack strategy card is the best one I’ve seen. Is it available any where? Theone I’m currently using does’t have surrender on it, what am I loosing by using this?

John C from Crestwood

Thanks. Almost every casino gift shop sells basic strategy cards, but for some reason they don’t indicate when to surrender. There are not many situations to surrender, but those situations happen often, so I think they are worth memorizing. In a six-deck game surrender is worth 0.07% if the dealer stands on soft 17, and 0.09% if he hits it.

Just thought you might find this interesting. At Bodog they offer the following bet, “Will Britney Spears and Kevin Federline's second baby be a Boy or Girl?” The odds on a boy are +105, and for a girl −145. Last I checked this has been 1:1 since the beginning of mankind. I'd like to know who is taking the -145 side of this one. Love the site, visit often, and click your sponsors in appreciation.

Anonymous

Thanks for the kind words. To be honest nobody cares much about click-through rates any longer. So don't feel obligated to click through the banners if it is just for show. To answer your question, in the United States the probabilities are very close to 50.5% boy and 49.5% girl. Assuming no other information is known by the betting community the player advantage on the boy bet would be .505*1.05 - .495 = 3.53%. It could be that somebody with inside knowledge is betting on a girl. Another theory is that some people incorrectly believe you can tell the gender by the shape of the mother’s belly, and these people are betting on a girl. Personally I’m going to leave this one alone.

Why isn’t "Thou shalt not cheat" in your 10 Commandments of Gambling? Your definition of Proper Gambling Etiquette doesn’t seem to include that definition.

Kevin from San Diego, CA

You’re right. That should be in there, and close to the top. I’ll add that in and add the commandment about tipping to the one about etiquette.

Anonymous

A word from the Wizard on this 200th "Ask the Wizard" column.

Welcome to my 200th “Ask the Wizard” column. I had no idea when I started this column on February 5, 2000, that I would get this far. It took four years and two months to get to the 100th column, and another three years and nine months to this one.

These columns may not seem like much, but a great deal of my time goes into them. It isn’t easy opening the door to questions for the column, without the whole world flooding me with requests for math tutoring, deconstruction of past gambling trips, feedback on worthless betting systems, general gambling questions for those too lazy to search the site, advice on alleged cheating boyfriends, and entire life stories.

Unfortunately, I recently decided to close the doors to new questions. When I clear out the existing backlog I will temporarily open them again. In the future, I plan to have open seasons for questions from time to time. I hope those with good questions will hang onto them until the next open season, and those with bad questions will not be so patient.

Some of the questions that do make the column take a lot of time to answer. For example, my last comment in this column about the effect of separate decks for the player and dealer took hours of time to reprogram my blackjack simulator to handle. Other numbers have had much more time and effort go into them. On average, it takes about two days to write each column.

In April 2005 I was so fatigued from gambling questions I stopped the column for an indefinite period of time, perhaps forever. However there was a loud outcry to bring it back, so in August I did. As I write this I am going through a bad case of “Ask the Wizard” fatigue again. Rather than stop cold turkey again, I plan to cut down on the frequency of the column, to just one or two a month.

I would like to take a moment to thank my proofreader, Don Schlesinger for his invaluable help correcting my bad English, over the last several months. Not only has it made the column much nicer to read, but he has helped my own grammar in everything I write.

I would also like to thank my readers for your support over the years. Ultimately, it is the readers that put rice on my table and send my kids to private school. I hope that in return I have helped you win, or at least lose less, in the casinos, by trusting in math and not myths. Until next time, set your expectations high!

Hi there. I often spend down time playing your free craps game. Thank you, I think it is great. I know nothing about computer programming, and my question is about the dice outcomes. Are these completely random rolls every time? Would you see similar outcomes of 100 consecutive rolls on the computer program as you would throwing dice 100 times?

Jordan

Thank you for the compliment. That program was written in JavaScript. Drawing a random number is a simple one line command. Math.random() will return a random decimal between 0 and 1. Here is a simple program to generate 25 dice throws. If you refresh this page, you should get different results every time.

Results

In my opinion, the random numbers generated are good enough for the purpose of the game. However, if I were making a real money game, I would take efforts to use a better, and more secure, random number generator.

Hello, I am a seventh grader from Hawaii. I am doing a science fair project on poker and shuffling. I am using your charts about the win percentage of pocket Texas Hold ’Em hands, according to the number of players in the game. I was hoping you could answer a few questions that would help me with my project:

  1. How did you come up with the percentages found in the charts?
  2. If you used a computer program, how did you develop it and how long did it take?
  3. You stated that you started the Wizard of Odds as a hobby. Did experimenting change as your site became more well-known? Why or why not?

    Anonymous

    1. The two-player table was done by a brute-force looping program, that cycled through all 1225 possible opponent cards, and 1,712,304 possible community cards. For three to eight players, looping would have taken a prohibitive amount of time, so I did a random simulation.
    2. I write almost all my programs in C++, including both programs I just mentioned. The rest are in Java or PERL. I mostly copied and pasted code from other poker-based programs. The new code only look about a day to write.
    3. Yes, I started my site as a hobby in June 1997. It wasn’t until January 2000 that I accepted advertising, and tried to make a business out of it. It has gone through three different domains over the years. Here is what it looked like in May 1999. The purpose of the site has always remained the same, a resource for mathematically-based gambling strategy. Through the years, I have just been adding more games and material. One experiment was providing my NFL picks for the 2005 season, which was an abject failure.

While playing your free blackjack game on this site, I noticed that when the count got to, i.e. -7 or lower, I was winning as much if not more than when the count was say +10 or higher. Is this just a fluke, or is there some advantage to the player when the count is a high negative number?

Mark from Pahokee

The game shuffles after every hand, so the count is always zero at the start of the hand. However, when there is a cut card, a negative count tends to be very slightly correlated to positive results since the start of the shoe. That is because the negative count means that more good cards than bad have already been played, which would have been good for the player.

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