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Ask The Wizard #306
At the Suncoast, there's an exciting poker promotion that rewards players with between $50 and $100 if they hold a specific high pocket pair but end up losing in Texas Hold 'Em. The ideal pocket pair varies daily, and may include jacks, queens, kings, or aces. Depending on the color of the cards in hand, a player can win $100 if both are black, $75 if they are both red, and $50 if one is black and one is red. How much is this promotion worth in terms of hourly value?
The value of the promotion is influenced by the number of participants at the table; the more players, the better your chances of losing, which activates the reward. A detailed table illustrates the probabilities of losing with each high pocket pair based on the total number of players at the table, assuming no one folds. While this assumption may not be reflective of an actual game, it serves as an upper limit for these probabilities.
Likelihood of Losing with High Pocket Pairs in Texas Hold 'Em
| Players | Jacks | Queens | Kings | Aces |
|---|---|---|---|---|
| 10 | 80.16% | 77.34% | 73.57% | 68.64% |
| 8 | 74.87% | 71.29% | 66.74% | 60.95% |
| 6 | 65.95% | 61.70% | 56.68% | 50.49% |
| 4 | 50.37% | 46.09% | 41.41% | 35.82% |
| 3 | 38.43% | 34.71% | 30.79% | 21.22% |
| 2 | 22.85% | 20.37% | 17.88% | 15.07% |
The expected average payout is calculated easily: $100 multiplied by 1/6, plus $75 multiplied by 1/6, plus $50 multiplied by 1/2, resulting in an average win of $62.50. The upcoming table outlines the expected value for each high pocket pair assuming that no one else folds during play.
Expected Win per Occasion
| Players | Jacks | Queens | Kings | Aces |
|---|---|---|---|---|
| 10 | $50.10 | $48.34 | $45.98 | $42.90 |
| 8 | $46.79 | $44.56 | $41.71 | $38.09 |
| 6 | $41.22 | $38.56 | $35.43 | $31.56 |
| 4 | $31.48 | $28.81 | $25.88 | $22.39 |
| 3 | $24.02 | $21.69 | $19.24 | $13.26 |
| 2 | $14.28 | $12.73 | $11.18 | $9.42 |
The following table provides insight into the worth of this promotion for each hand played. It calculates this by multiplying the expected winnings by the chance of drawing the required hole cards, which is computed as 6 out of 1326 or approximately 0.90%.
Expected Win per Hand Played
| Players | Jacks | Queens | Kings | Aces |
|---|---|---|---|---|
| 10 | $0.23 | $0.22 | $0.21 | $0.19 |
| 8 | $0.21 | $0.20 | $0.19 | $0.17 |
| 6 | $0.19 | $0.17 | $0.16 | $0.14 |
| 4 | $0.14 | $0.13 | $0.12 | $0.10 |
| 3 | $0.11 | $0.10 | $0.09 | $0.06 |
| 2 | $0.06 | $0.06 | $0.05 | $0.04 |
The next table reflects the value of this promotion calculated on an hourly basis, presuming an average play rate of 30 hands per hour. Again, this scenario works on the assumption that none of the players will fold, establishing an upper limit on the hourly value.
Expected Win per Hour Played
| Players | Jacks | Queens | Kings | Aces |
|---|---|---|---|---|
| 10 | $6.80 | $6.56 | $6.24 | $5.82 |
| 8 | $6.35 | $6.05 | $5.66 | $5.17 |
| 6 | $5.60 | $5.23 | $4.81 | $4.28 |
| 4 | $4.27 | $3.91 | $3.51 | $3.04 |
| 3 | $3.26 | $2.94 | $2.61 | $1.80 |
| 2 | $1.94 | $1.73 | $1.52 | $1.28 |
Let's consider a game that operates under these specific rules:
- A random number generator recreates numbers uniformly distributed between 0 and 1.
- Two players each receive distinct numbers that only they can view.
- Player 1 has the choice to either keep his original number or opt for a new random one.
- Knowing Player 1's choice, Player 2 faces the same dilemma: maintain his original number or switch for a new random value.
- Player with the higher number wins.
I have four questions about the game:
- Address the subsequent queries related to the game:
- What value would make Player 1 indifferent between sticking with his number or changing it?
- If Player 1 decides to switch, at which number would Player 2 feel indifferent about changing his number?
- If Player 1 chooses to remain with his initial number, what value would make Player 2 ambivalent towards switching?
- Assuming that both players are employing optimal strategies, what are the odds that Player 1 will emerge victorious?
The answer and the methodology can be found on my designated page for this topic. Math Problems , problem 225.
Is selecting the 'Jackpot Only' option in the Mega Millions lottery a worthwhile choice?
Disregarding taxes, annuity payouts for the jackpot, and sharing of the jackpot, you should opt for the 'Jackpot Only' choice only if the jackpot exceeds $224,191,728. Taking those elements into account, the recommended approach would be to consistently choose the Megaplier instead.
For a more detailed discussion, please refer to my dedicated page. Mega Millions lottery .
In the Sky City casino based in Auckland, New Zealand, it is required for both the dealer and player to utilize both of their hole cards in the game of Ultimate Texas Hold 'Em. How does this stipulation alter the odds from standard rules where any combination of five cards can be used?
This regulation raises the house edge from 2.19% to 7.97% and increases the Element of Risk from 0.53% to 1.90%. The reasons for this change include the dealer's inability to qualify as often, and the challenge of winning on the Blind bet, which necessitates a straight or better.
For an in-depth look at my analysis, check out my latest page discussing the subject. Auckland's variation of Ultimate Texas Hold 'Em .
For further dialogue on this topic, please visit the discussion thread. ULTIMATE IN NEW ZEALAND in my forum at Wizard of Vegas.