Ask The Wizard #54

I have noticed something over the years watching our Quinto lottery drawing here in Washington State. It is a 52 ’card’ game that draws 5. I have noticed that the vast majority of the time, three suits are drawn. From poker numbers you find that the chance of getting just one suit (a flush) is 5148 out of about 2.6 million. What are the odds of 2,3, or all 4 suits coming up?

Kevin from Tacoma, Washington

Let’s define f(x,y) as the probability of getting x of one suit and y of another. This function is not limited to two terms.

With two arguments f(x,y)= combin(13,x)*combin(13,y)*12/combin(52,5).

With three arguments f(x,y,z)= combin(13,x)*combin(13,y)*combin(13,z)*12/combin(52,5).

With four arguments f(w,x,y,z)=combin(13,w)*combin(13,x)*combin(13,y)*combin(13,z)*4/combin(52,5).

The probability of all four suits is COMBIN(13,1)3*COMBIN(13,2)*4/combin(52,5) = 26.37%.

The probability of three suits is COMBIN(13,3)*COMBIN(13,1)2*12 + COMBIN(13,1)*COMBIN(13,2)^2*12/combin(52,5) = 58.84%

The probability of two suits is COMBIN(13,3)*COMBIN(13,2)*12 + COMBIN(13,4)*COMBIN(13,1)*12/combin(52,5) = 14.59%

The probability of one suit (including straight and royal flushes ) is 4*combin(13,5)/combin(52,5) = 0.20%.

So three suits are the most frequent outcome.

What is the best game to use a match play coupon on?


First let’s define a match play coupon for those who don’t know. This is something often found in casino fun books. If the player accompanies a match play coupon with a real even money wager then the match play will be converted to a like amount of cash if the player wins. For example if the player has a $5 match play and uses it along with a $5 bet on red in roulette then if the player wins his $5 will win $5 and his match play will be converted to $5. Whether the player wins or loses he will lose the match play coupon. In the event of a push, the player gets to keep the match play coupon.

If used in blackjack, the Match Play will usually only pay even money. This decreases the value of the Match Play itself by 2.3%, which is way too much. Of the true even money bets, the best game to use a match play on in the Player bet in baccarat. That has a probability of winning of 49.32% of bets resolved. For the don’t pass in craps, that probability is 49.30%.The value of a Match Play on the Player bet is 47.95% of face value, assuming you wouldn’t have bet otherwise.

On a CSM blackjack game, 5 deck, what would be the effect of the dealer not putting discards back into the machine every hand if 24 of 52 cards in the discard rack were face cards? What about 48 of 102? What would be the effect if 44 of 52 cards were non-face cards? Can the odds be heavily skewed? I have a feeling that the odds skyrocket in the casino’s favor if the dealer leaves face cards in the discard.

Doug from Vancouver, Canada

The exact numbers would be difficult to calculate and I won’t get into that. However your speculation is right that the odds favor the dealer if he leaves a lot of high cards in the discard rack yet will put back into play a lot of small cards. This would be the same kind of thing as preferential shuffling, in which the dealer of a hand held game shuffles when the count is good but deals another round on a bad count. Preferential shuffling is something that definitely does happen here in Las Vegas so what you describe would not surprise me either.

Can you please explain the "cut card effect" in Blackjack. Why don’t people use this when quoting BS house edges for various numbers of decks rather than showing BS house edges that take this into account?

Andrew from Melbourne, Australia

I explain all about the cut card effect in my blackjack appendix 10. The reason house edge statistics are based on a non-cut card game is likely out of tradition and mathematical simplicity. However you make a good point that it would be more accurate to inflate the house edge figures to account for the cut card. Yet I am still going to stick to basing my house edge figures on the first hand of a freshly shuffled deck. Blackjack is hard enough to analyze without worrying about the cut card effect all the time.

I have seen several references to ongoing bonus abuses from people who reside in Denmark. Do you know more details about what is going on here? What are they doing, and how can we avoid being bonus abusers?


I don’t know all the details but there is a lot of bonus abuse going on by players from Denmark. The way to avoid being labeled a bonus abuser is to always play much more than required. I hate to give an exact figure but exceeding requirements by at least 100% is a good idea. Giving free play to casinos you get repeated bonuses from is also good camouflage. It doesn’t look like you are a good faith gambler if you only play during promotions. In general don’t be too greedy.

I’ve got a question about "progressive betting" (e.g. "Another Experiment", Player 2 on your Betting Strategies page). Obviously in normal bj play you experience streaks of wins and loses. Where is the faulty logic in "minimize your losing streaks by resetting at 1 unit, and increase your winning streaks by raising 1 unit after each win?" FYI, I actually play a little variation of that: 15, 30, 45, 50, 75, 100, 125, etc...Thanks for you time. And, please don’t try to humiliate me like Ann {what’s-her-name} on The Weakest Link :-) I really love your site!!! Thanks for all of the great info.

Chad R. from Memphis, Tennessee

Progressive betting systems, like yours, will turn a good session into a great one without the risk of catastrophic loss as with regressive systems like the Martingale. However progressive systems will turn a choppy neutral session into a bad one. Consider what would happen if you alternated between a win and a loss the entire session. The wins would all be at $15 and the losses at $30. Funny you should mention the ’Weakest Link.’ I tried out for that show during the summer and didn’t make it. It is probably just as well because I’m not that witty in real life and doubt I could come up with a good rejoinder to one of Ann’s jabs.