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Ask the Wizard #117

What is your advice for playing rocks/paper/scissors?

"Anonymous" .

The best piece of advice anywhere on this site may be this: The first round, ALWAYS PICK PAPER. That is because amateur players tend to pick rock the first time. Just hold out your hand in each position, one at a time, and you’ll see that rock is the most comfortable and natural choice. If you play repeated rounds you should pick whatever would beat your opponent the last round with probability less than one-third. This is because I believe amateurs repeat less than one-third of the time. If playing a pro who you fear can get into your head then randomize by looking at the second hand of your watch, divide the number of seconds by three and take the remainder, then map the remaider as follows 0=rock, 1=scissors, 2=paper (or any other mapping as long as determined in advance). So the next time you go to a restaurant Dutch style I suggest playing a single round for the check and then pick paper. You can thank me later.

In Las Vegas the majority of the casinos I’ve played in require a $100 blackjack bet for 4 hours daily to get a room comped. I’ve calculated that playing with S17,RSA,DOA,LSR assuming 100 hands per hour it is costing me $120 per day if I play basic strategy. Is there another game that I would lose less if I played? I’m mainly concerned with the cost of Pai Gow if I never bank playing $100 a hand 4 hours a day. However, I thought you might know of another game it would be even cheaper for me to play.

Jay F.

The rule of thumb when it comes to comps is that the casinos give back some percentage, usually one-third. So if your goal is to get the room with as little expected loss as possible then whatever game offers the lowest house edge is what you should play. You will probably earn that room faster and with less bankroll volatility playing pai gow or pai gow poker. However the house edge is higher so your expected loss will be greater than in blackjack. In my opinion you should play whatever you would play if there were no comps at all. Then consider comps as icing on the cake.

The casino industry has millions of customers, wagering and losing billions of dollars, and a percentage of them unhappy enough to claim that they were cheated by lying advertising. Do you think the trial lawyers are a definite threat to do to this industry after what they have done to asbestos, tobacco and others?

"Anonymous" .

I hope not. If anyone brings such a lawsuit I hope the casino wins. As long as the casinos are operating honestly and fairly, which in general I believe they are, then if the player loses more than he can afford it is his own fault. I’m not a lawyer but nobody here in Vegas seems very worried about this.

I ran into a situation in Reno a couple years ago that no one else seems to have heard before. At a full pai gow poker table, the dealer set her cards, I believe it was a Jack/Ten in the 2-card hand and a flush in the 5 card-hand. The dealer had missed the fact there was a straight with a higher 2-card hand, but set it and went through 4 players when the pit boss came over and said "You set that wrong" and proceeded to re-set the hand. They then went to the discard tray to replay the hands. This resulted in two of the players going from a push to a loss. The pit boss actually went to the players’ stacks and took the monies from them after consulting the videotape to confirm the wager amounts. We were all told to stay at the table until the situation was resolved, but after it was, despite not knowing each other, we all left not only the table, but the casino entirely. It would strike me that once a hand is set and the first hand is settled, there can be no change. Also, for PR purposes, that pit boss lost my business forever over a mess that netted the casino $20. What do you think?

Kevin H.

The casino had the right to do this. However in my opinion it was a bad business decision. Not only did the casino waste time resolving this mess but as you point out it resulted in bad feelings on the part of all players. This just goes to show the folly of following rules religiously. Personally I think rules should be weighed against common sense.

I am a casual Basic Strategy blackjack player who would like to learn a counting technique that offers positive or even expected value. I was wondering if there is a counting technique that doesn’t require tracking all the cards. I’m thinking that I could take a random sample of all the cards played (say my hand) and just count the cards that have been in my hand. Hopefully, my cards are representative of all cards played and thus the count of my hand is close enough to the true running count that I can adjust my wagering accordingly. Do you feel that this could be a valid approach particularly against 6-8 deck shoes or am I just a lazy guy looking for the easy way out?

"Anonymous" .

The easiest counting method is what I call the "eyeball" method, in which if you see lots of small cards come out then increase your bet, and vise versa. However this is better suited to one and two deck games. For 6 or 8 deck games I would recommend the ace/five count. This requires only counting aces and fives. According to Ken Uston in Million Dollar Blackjack this give the player an additional 0.5% with just a 1 to 3 unit bet range. That is enough to overcome the house edge in most games.

I recently attended a hospital fete. There was a new car as a prize if 7 dice produced 7 sixes in one throw. £1.00 a go. Odds on this must be high but how high?

"Anonymous" .

The probability of throwing seven sixes with seven dice is (1/6)7 = 1 in 279,936. So the car would have to have a value of £279,936 or more for this to be a good bet. Even your average Rolls Royce is not worth this much, so I would say that was a terrible bet.

[Bluejay adds: Uh, yeah, but I think the point was that it was for charity. What’s more fun: Donating £1.00 to charity and getting nothing back but the good feeling of helping out, or donating £1.00 and getting the good feeling plus the longshot chance of winning a car?]

Dear Wizard, I know from reading your web site and from other sources that betting systems do not give you an advantage over the house. My question is do they decrease the house advantage? I have been playing the betting system outlined in Progression Blackjack by Donald Dahl for the last 8 years and it gives me the excitement of betting higher amounts then I normally would. I usually play the $10 tables and I often get up to $30 bets and on my last trip to Vegas I got up to the $100 level at Sam’s Town which really got my heart pumping not to mention a $600 profit that I left the table with. Thanks for your help.


No! Not only do betting systems not overcome the house edge but they can’t even put a dent in it. Nor can they increase the house edge. All they can do is affect volatility. Since it sounds like you like a volatile, exciting game than your system is fulfilling its purpose. Just don’t expect to win.

What is the probability, over the course of 1 million hands, that there is a royal flush drought extending for 200,000 hands? I'm more interested in the solution than the answer itself.

"Anonymous" .

It isn't often I say this but I tried for hours but the math on this one was simply over my head. So I turned to my friend and math professor Gabor Megyesi. Here is his formula for any "drought" problem.

  1. Let p be the probability of winning any given hand.
  2. Let d be the length of the drought.
  3. Let n be the number of hands played.
  4. Set k=dp and x=np.
  5. If k=1 then let a=-1, otherwise find a such that k=-ln(-a)/(1+a). (a is a negative number, if k>1 then -1 < a < 0, if k < 1 then a < -1, and a needs to be calculated to high accuracy.) [Wizard’s note: This kind of solution can be easily found in Excel using the Goal Seek feature under the tools menu.]
  6. if k=1 then let A=2, otherwise let A=(1+a)/(1+ak).
  7. The probability of no drought of length d in n hands is approximately Aeax.

In this particular problem p=1/40391, d=200000, n=1000000, k=4.9516, x=24.758, a=-0.0073337, A=1.03007. So the probability of no drought is 1.03007*e-0.0073337*24.758 = 0.859042. Thus the probability of at least one drought is 1-0.859042 = 0.140958.

Here is Gabor Megyesi's full 5-page solution (PDF). Thanks Gábor for your help.

I did a random simulation of 32,095 sets of one million hands. The number with at least one drought was 4558, for a probability of 14.20%.

I play occasionally with a group of players who love poker but occasionally want to play BJ to vary the evenings proceedings. Most of them would be beginners in terms of strategy and probability awareness. What would be a fair set of rules you would recommend so that BJ becomes a fair game (or as close as possible) for both players and whoever takes the bank?

"Anonymous" .

It would depend on the specific skill factor of the players. Without knowing that, but assuming the skill level is equal among players, I would have the bank option rotate from player to player.

Why is it better odds for the casino to hit on a soft 17? It seems they would be more likely to bust and hence have worse odds.

"Anonymous" .

It is true the casino busts more often if the dealer hits a soft 17. However the dealer also gets fewer seventeens, which is not a very good hand. It is to the dealer’s advantage to hit a soft 17 for the same reason the player should always hit or double on a soft 17. A 17 is a lousy hand, and whether the player or the dealer hitting a soft 17 offers two chances to improve upon it.

What are the probabilities for a 5 of a kind, 4 of a kind, 3 of a kind, full house, 2 pair, pair, straight, and nothing with the roll of five dice?

"Anonymous" .

  • Five of a kind: 6/65 = 0.08% (obvious)
  • Four of a kind: 5*6*5 = 1.93% (five possible positions for the singleton * 6 ranks for the four of a kind * 5 ranks for the singleton).
  • Full house: combin(5,3)*6*5/65 = 3.86% (combin(5,3) positions for the three of a kind * 6 ranks for the three of a kind * 2 ranks for the pair).
  • Three of a kind: COMBIN(5,3)*COMBIN(2,1)*6*COMBIN(5,2) / 65 = 15.43%. (combin(5,3) positions for the three of a kind * combin(2,1) positions for the larger of the singletons * 6 ranks of the three of a kind * combin(5,2) ranks for the two singletons.
  • Two pair: COMBIN(5,2)*COMBIN(3,2)*COMBIN(6,2)*4 / 65 = 23.15% (combin(5,2) positions for the higher pair * combin(3,2) positions for the lower pair * combin(6,4) ranks for the two pair * 4 ranks for the singleton.
  • Pair: COMBIN(5,2)*fact(3)*6*combin(5,3) / 65 = 46.30% (combin(5,2) positions for the pair * fact(3) positions for the three singletons * 6 ranks for the pair * combin(5,3) ranks for the singletons.
  • Straight: 2*fact(5) / 65 = 3.09% (2 spans for the straight {1-5 or 2-6} * fact(5) ways to arrange the order).
  • Nothing: ((COMBIN(6,5)-2)*FACT(5)) / 65 = 6.17% (combin(6,5) ways to choose 5 ranks out of six, less 2 for the straights, * fact(5) ways to arrange the order.

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.