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Ask the Wizard #287
"Anonymous" .
For the benefit of other readers, a Yahtzee is a five of a kind with five dice. In the game of Yahtzee the player may hold any dice he wishes and reroll the rest. He can do this up to three rolls.
The player may reroll previously held dice, if he wishes. For example, if the player's first roll is 33456 and he holds the threes and then has 33555 after the second roll he may keep the fives and reroll the threes on his third roll.
The following table shows the maximum number of dice of the same face for 1 to 20 rolls. The table shows the probability of getting a Yahtzee within three rolls is about 4.6%.
Yahtzee Probabilities
Rolls  Maximum Dice of Same Face  

One  Two  Three  Four  Five  
1  0.092593  0.694444  0.192901  0.019290  0.000772 
2  0.008573  0.450103  0.409022  0.119670  0.012631 
3  0.000794  0.256011  0.452402  0.244765  0.046029 
4  0.000074  0.142780  0.409140  0.347432  0.100575 
5  0.000007  0.079373  0.337020  0.413093  0.170507 
6  0.000001  0.044101  0.263441  0.443373  0.249085 
7  0.000000  0.024501  0.199279  0.445718  0.330502 
8  0.000000  0.013612  0.147462  0.428488  0.410438 
9  0.000000  0.007562  0.107446  0.398981  0.486011 
10  0.000000  0.004201  0.077416  0.362855  0.555528 
11  0.000000  0.002334  0.055317  0.324175  0.618174 
12  0.000000  0.001297  0.039279  0.285674  0.673750 
13  0.000000  0.000720  0.027757  0.249063  0.722460 
14  0.000000  0.000400  0.019543  0.215313  0.764744 
15  0.000000  0.000222  0.013720  0.184883  0.801175 
16  0.000000  0.000124  0.009610  0.157896  0.832371 
17  0.000000  0.000069  0.006719  0.134258  0.858954 
18  0.000000  0.000038  0.004692  0.113753  0.881517 
19  0.000000  0.000021  0.003272  0.096100  0.900607 
20  0.000000  0.000012  0.002280  0.080994  0.916714 
This question is raised and discussed in my forum at Wizard of Vegas.
cwwbjr
It depends on the game, naturally. The greater the house advantage and lesser the variance, the greater the chance is. It also depends on the level of confidence.
The following table shows how many bets are required, assuming the same bet amount, for the casino to be confident, at various levels of confidence, for some common games. For example, to have a 95% chance of showing a net profit on the Banker bet in baccarat, the casino would need to deal 20,791 hands.
This is based on the Normal Distribution in all cases except for Jacks or Better. That approximation becomes untrustworthy if the number of expected events of any one outcome is five or less. So, for video poker, I used the Poisson distribution for the royals and the Normal approximation otherwise.
For blackjack, the rules are: 6 decks, dealer stands on soft 17, double after split allowed, surrender allowed, resplitting aces allowed.
This question is raised and discussed in my forum at Wizard of Vegas.
rjs357
For the benefit of other readers, here is the return table for 96 Jacks or Better.
"96" Jacks or Better
Hand  Payoff  Combinations  Probability  Return 

Royal flush  800  493512264  0.00002476  0.01980661 
Straight flush  50  2178883296  0.00010931  0.00546545 
Four of a kind  25  47093167764  0.00236255  0.05906364 
Full house  9  229475482596  0.01151221  0.10360987 
Flush  6  219554786160  0.01101451  0.06608707 
Straight  4  223837565784  0.01122937  0.04491747 
Three of a kind  3  1484003070324  0.07444870  0.22334610 
Two pair  2  2576946164148  0.12927890  0.25855780 
Jacks or Better  1  4277372890968  0.21458503  0.21458503 
Nothing  0  10872274993896  0.54543467  0 
Total  19933230517200  1  0.99543904 
Most of my video poker return tables for 52card games have the same number of combinations of 19933230517200. The question is why?
First, there are combin(52,5) = 2,598,960 ways to choose five cards out of 52.
Second, there are up to combin(47,5) = 1,533,939 combinations on the draw, depending on how many cards the player discards. The following table shows the number of draw combinations in the second column according to the number of discards.
Draw Combinations
Discard  Combinations  Weight  Product 

0  1  7,669,695  7,669,695 
1  47  163,185  7,669,695 
2  1,081  7,095  7,669,695 
3  16,215  473  7,669,695 
4  178,365  43  7,669,695 
5  1,533,939  5  7,669,695 
The least common multiple of the numbers in the second column is 7,669,695. This number can be expressed as 5×combin(47,5). To keep the total number of combinations the same for every hand on the deal I weight the draw combinations so that the total combinations on the draw is 7,669,695.
So, 19,933,230,517,200 = combin(52,5)×combin(47,5)×5. Some of my 52card video poker tables have a smaller number of combinations. This is because sometimes the total number of combinations for each hand in the final return table has a greatest common divisor greater than one. In this case I sometimes divide each total by whatever the greatest common divisor is. My video poker analyzer does this automatically.
This question was raised and discussed at my Wizard of Vegas forum.
Joe Shipman from New Jersey
They showed me the house way and it did not clearly address this hand. They ended up taking my money and giving me double my bet in nonnegotiable chips so that the game could continue, which was okay. There was a pow wow in the pit with 4+ suits about how to set the hand and it lasted well over half an hour. They finally decided that TTTJK/A* was the proper way to set it and would change the house way as such.
My question is what is the house way for this hand and how do you interpret the house way for confusing hands like this?
bigfoot66
Pai gow poker was a badly designed game from day one and decades later nobody has bothered to clean it up. Flaws to pai gow poker include:
 An overly confusing and complicated house way.
 The ridiculous rule that the A2345 straight is the second highest.
 Option for player banking and cobanking, which almost nobody, except me, invokes.
When I briefly worked for a major Strip casino, that shall remain nameless, I offered to create a house way that would have been shorter, more powerful, and clearly covered every possible situation. Not just for pai gow poker, but tiles as well. Of course, that suggestion was shot down without comment.
Now that my rant is out of the way, I'll try to answer your question. While there are some differences from one house way to another, they are all more or less organized the same way. The Foxwoods house way is typical. Using your hand as an example, it isn't clear whether to treat is as a straight hand and play it as AKQJT/TT or a full house hand and play it as TTTJK/AA.
Not that you asked, but let's look at what would be the mathematical better play for the house. Using my pai gow poker appendix 1 we get the following, assuming the dealer is banking:
By playing the hand as a straight the dealer can expect to win 96.46% of the money, as opposed to 87.49% as a full house. So, going by the straight rule is significantly better.
Personally, the way I code pai gow poker is to start with the highest ranked hand (five aces) and move my way up the page in an if/else if/else if/else if/end loop. In other words, I classify the hand according to the highest possible fivecard hand that can be made and then follow those rules.
I think that this is how the house way is intended to be interpreted. As evidence why, consider the Canterbury Park house way. In particular the rules for flush hands, which reads as follows.
Six and seven card flushes play the highest two cards possible in the Low Hand.
(Exception: If two pair are with the flush, the two pair rule will apply.)
As with all other house ways, the lowest hands are listed first. If the hand was supposed to be played by the first set of rules encountered, then they would have followed the two pair rule anyway. There would be no reason to explicitly make an exception to follow the two pair rule, since it was listed before the flush rule.
For practical purposes, I asked two dealers how they would handle this hand. They both pretty much said that the written house way is just a guideline and that in the event of a confusing hand, ask the floor what to do, and then just do as you're told without asking why. One of the dealers said that an unspoken policy is to go by the rule that seems to fit that situation best. In the case of Foxwoods, there is a rule for straight hands with a three of a kind. Since that most specifically describes the hand in question, follow that rule, despite the fact that the full house rule covers it too.
To make a long story short, no casino house way I have ever seen, in either pai gow poker or pai gow tiles, clearly explains how to play every possible hand. They are full of contradicting rules. Until somebody cares to do anything about it, you'll have to go by the interpretation of whoever is working the floor. In my experience, these confusing cases seem to usually get ruled against the player.
I hope one of these days a player will get fed up with this and make a complaint to the appropriate Gaming authority the next time an ambiguous hand is set in a way that favors the dealer.
This question is raised and discussed in my forum at Wizard of Vegas.