Ask The Wizard #287

To clarify for others, a Yahtzee refers to rolling five matching dice in the game. Players can choose which dice to set aside and re-roll the others, with a maximum of three rolls allowed.

Players can opt to re-roll any dice they have previously set aside. For instance, if the first roll results in 3-3-4-5-6 and the player holds onto the threes, they could attempt to re-roll those for better results on their third roll after obtaining 3-3-5-5-5 in their second attempt.

The table below illustrates the highest number of matching dice faces achievable from 1 to 20 rolls. It indicates that your chances of rolling a Yahtzee within three attempts are approximately 4.6%.

This topic is brought up and discussed in my community forum at Wizard of Vegas .

This varies based on the game itself. Generally, when the house edge is higher and the outcome unpredictability is lower, the chances of profitability rise. Additionally, it depends on the desired level of certainty.

The following table details the number of plays needed, based on consistent betting amounts, for casinos to achieve confidence at different confidence intervals across various popular games. For instance, to attain a 95% probability of generating a profit on the Banker bet in baccarat, 20,791 hands would need to be played.

This analysis relies on the Normal Distribution for all scenarios except for Jacks or Better . This estimation loses credibility when the expected count of any individual outcome is five or fewer. Therefore, I employed the Poisson distribution for royal flushes in video poker and utilized the Normal approximation in other situations.

For blackjack, the following conditions apply: using 6 decks, the dealer stands on a soft 17, players can double down after splitting, and the options to surrender and re-split aces are available.

This topic is brought up and discussed in my community forum at Wizard of Vegas .

For the benefit of others, here is the return table for the 9-6 version of Jacks or Better.

Most of my return tables for video poker with a standard 52-card deck exhibit a common combination count of 19933230517200. What’s behind this?

First, there are combin (52,5) = 2,598,960 ways to choose five cards out of 52.

Additionally, depending on how many cards the player discards, there can be as many as combin(47,5) = 1,533,939 possible combinations on each draw. The next table displays the number of combinations available for drawing according to the count of discarded cards.



The least common multiple of the figures listed in the second column is 7,669,695. It can be represented as 5×combin(47,5). I balance the draw combinations so that the overall possible combinations during the draw remain at 7,669,695.

So, 19,933,230,517,200 = combin(52,5)×combin(47,5)×5. Some of my video poker return tables show fewer combinations because the total combinations for each hand in the final return table may have a greatest common divisor that exceeds one. In such cases, I sometimes simplify by dividing each total by this divisor. video poker analyzer does this automatically.

This subject was addressed and explored in my Wizard of Vegas forum.

Good question! Here is my answer and cursory solution . Also see my solution in PDF form .

From inception, Pai Gow Poker has shown inherent design flaws, and years later, little has been done to rectify the complexities. Among its problematic elements are:

• An excessively complicated house way that confuses players.
• The absurd ruling that A2345 constitutes the second-highest straight.
• An option for player banking and co-banking that is rarely exercised, only by players like myself.

During a brief tenure at an unnamed major Strip casino, I proposed a simplified and more powerfully defined house way that would comprehensively cover all scenarios—not just for pai gow poker but for tiles as well. Naturally, my suggestion was dismissed without feedback.

Having expressed my frustrations, I will now aim to address your inquiry. While variations exist across different house ways, they generally follow a similar structure. The Foxwoods house way is commonly adopted. In reference to your hand, the decision regarding whether to treat it as a straight with AKQJT/TT or a full house with TTTJK/AA isn’t immediately clear.

Although you didn’t solicit my input, let’s analyze what would mathematically benefit the house. With my pai gow poker appendix 1 we ascertain that if the dealer acts as the banker:



By treating the hand as a straight, the dealer can anticipate winning 96.46% of the wagers, compared to just 87.49% as a full house. Opting for the straight is evidently more advantageous.

In my practice of programming pai gow poker, I start by identifying the highest-ranked hand (five aces) and methodically progress through an if/else if structure. Essentially, I categorize the hand based on the strongest five-card hand possible and then adhere to those rules.

I believe this approach aligns with the intended interpretation of the house way. To illustrate my point, consider the Canterbury Park house way regulations. Particularly, the guidelines concerning flush hands stipulate:

For flushes with six or seven cards, the highest two cards are utilized for the Low Hand.
(Exception: If there are two pairs within the flush, the two pair rule takes precedence.)

Similar to other house methods, the lesser hands are listed first. Were the hand intended to be played according to the first set of rules presented, the two pair rule would have been applied without issue. There would be no rationale to introduce an exception favoring the two pair rule since it was stated prior to the flush guideline.

To gain practical insights, I consulted with two dealers regarding how they would approach this hand. Both indicated that the documented house way serves merely as a guideline, and in cases of ambiguity, it’s best to seek guidance from the floor staff and comply with their instructions without questioning their decisions. One dealer mentioned an unspoken guideline whereby one should adopt the rule that best suits the scenario at hand. In this situation, the rule for straight hands with three of a kind is most fitting to follow, despite the full house rule’s applicability. Foxwoods To summarize, I have yet to encounter a casino house way that definitively clarifies the proper play for all possible hands in either pai gow poker or pai gow tiles. They are riddled with conflicting directives. Until someone prioritizes improving this, you will invariably rely on the interpretation of the staff on duty. Based on my own experiences, these confusing scenarios seem to more often result in rulings that disadvantage the player.

I hope one day a frustrated player will escalate this matter to the appropriate Gaming authority if an ambiguous hand is resolved in ways that benefit the dealer.

Proven strategies and information for casino games such as blackjack, craps, roulette, and numerous others are available for enthusiastic players.

This topic is brought up and discussed in my community forum at Wizard of Vegas .