20 40 Stud Poker
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Introduction
I first noticed Big Raise Stud Poker at the Golden Nugget in Las Vegas on January 9, 2014. Before that it was known to be at the Venetian. It is a simple poker-based game based on the player's five-card hand only. It is similar to Mississippi Stud, but the player has one raise opportunity only, and can do so up to four times his Ante. Following are the complete rules.
Rules
- The Game is played with a single 52-card deck.
- The player makes an Ante bet.
- The dealer will deal each player three cards plus two community cards face down.
- The player may examine his own cards.
- The player has the option to fold or make a Raise bet. The Raise bet may be one to four times the Ante bet.
- The dealer will reveal the two community cards.
- Both Ante and Raise will pay according to the following pay table below.
Ante & Raise Pay Table
Hand Pays Royal flush 500 to 1 Straight flush 100 to 1 Four of a kind 40 to 1 Full house 8 to 1 Flush 6 to 1 Straight 4 to 1 Three of a kind 3 to 1 Two pair 2 to 1 High pair (10-ace) 1 to 1 Medium pair (6-9) Push All other Loss - There are also two side bets available, which I explain below.
Strategy
For purposes of the strategy I shall use the following notation:
- High card = 10 to ace.
- Medium card = 6 to 9.
- Low card = 2 to 5.
Make a 4x raise with:
- Three of a kind.
- Medium or high pair.
- Three to a royal flush.
- Three to a straight flush with no gaps, five-low or better.
- Three to a straight flush with one gap, six-low or better .
- Three to a straight flush with two gaps and at least two high cards.
Make a 1x raise with:
- Three to a flush.
- Low pair.
- All three cards medium or high.
- Two high cards and one low card.
- Three to a straight with one or less gaps and two medium cards.
Analysis
The following table shows all possible outcomes for the game. The lower right cell shows a house edge of 2.58%.
Big Raise Stud Poker — Return Table
| Hand | Raise | Pays | Combinations | Probability | Return |
|---|---|---|---|---|---|
| Royal flush | 4 | 2500 | 40 | 0.000002 | 0.003848 |
| Straight flush | 4 | 500 | 168 | 0.000006 | 0.003232 |
| Four of a kind | 4 | 200 | 5,088 | 0.000196 | 0.039154 |
| Full house | 4 | 40 | 27,072 | 0.001042 | 0.041666 |
| Flush | 4 | 30 | 5,012 | 0.000193 | 0.005785 |
| Straight | 4 | 20 | 3,120 | 0.000120 | 0.002401 |
| Three of a kind | 4 | 15 | 284,052 | 0.010929 | 0.163942 |
| Two pair | 4 | 10 | 516,348 | 0.019867 | 0.198675 |
| High pair (10-ace) | 4 | 5 | 1,295,304 | 0.049839 | 0.249197 |
| Medium pair (6-9) | 4 | 0 | 1,031,136 | 0.039675 | 0.000000 |
| Loser | 4 | -5 | 78,420 | 0.003017 | -0.015087 |
| Royal flush | 1 | 1000 | - | 0.000000 | 0.000000 |
| Straight flush | 1 | 200 | 192 | 0.000007 | 0.001478 |
| Four of a kind | 1 | 80 | 1,152 | 0.000044 | 0.003546 |
| Full house | 1 | 16 | 10,368 | 0.000399 | 0.006383 |
| Flush | 1 | 12 | 46,068 | 0.001773 | 0.021271 |
| Straight | 1 | 8 | 64,320 | 0.002475 | 0.019799 |
| Three of a kind | 1 | 6 | 179,748 | 0.006916 | 0.041497 |
| Two pair | 1 | 4 | 463,212 | 0.017823 | 0.071292 |
| High pair (10-ace) | 1 | 2 | 1,891,896 | 0.072794 | 0.145589 |
| Medium pair (6-9) | 1 | 0 | 1,135,584 | 0.043694 | 0.000000 |
| Loser | 1 | -2 | 7,802,820 | 0.300229 | -0.600457 |
| Fold | -1 | 11,148,480 | 0.428959 | -0.428959 | |
| Total | 25,989,600 | 1.000000 | -0.025751 | ||
On average, the player will make a 4x raise 12.49% of the time, a 1x raise 44.62%, and fold 42.90%. This makes the average final wager 1.95 units. Thus, the element of risk is 1.32%.
3 Card Bonus
The 3 Card Bonus is a side bet that pays based on the player's initial three cards only. Mathematically, it is exactly like the Pairplus bet in Three Card Poker.
The following return table shows a house edge of 7.28%.
3 Card Bonus
| Hand | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Straight flush | 40 | 48 | 0.002172 | 0.086878 |
| Three of a kind | 30 | 52 | 0.002353 | 0.070588 |
| Straight | 6 | 720 | 0.032579 | 0.195475 |
| Flush | 3 | 1096 | 0.049593 | 0.148778 |
| Pair | 1 | 3744 | 0.169412 | 0.169412 |
| Loser | -1 | 16440 | 0.743891 | -0.743891 |
| Total | 22100 | 1.000000 | -0.072760 |
Progressive Bet
The Progressive is $1 side bet based on the player's final five-card hand. A royal flush pays a progressive jackpot and a straight flush pays 10% of the jackpot. There are also envy bonuses if another player has a straight flush or higher.
The following table shows the return based on a hypothetical jackpot of $100,000. All wins are on 'for one' basis. The lower right cell shows a return of 82.30%, which equates to a house edge of 17.70%.
Progressive Bet
| Hand | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Royal flush | 100000 | 4 | 0.000002 | 0.153908 |
| Straight flush | 10000 | 36 | 0.000014 | 0.138517 |
| Four of a kind | 300 | 624 | 0.000240 | 0.072029 |
| Full house | 50 | 3,744 | 0.001441 | 0.072029 |
| Flush | 40 | 5,108 | 0.001965 | 0.078616 |
| Straight | 30 | 10,200 | 0.003925 | 0.117739 |
| Three of a kind | 9 | 54,912 | 0.021128 | 0.190156 |
| All other | 0 | 2,524,332 | 0.971285 | 0.000000 |
| Total | 2,598,960 | 1.000000 | 0.822994 |
Overall, the return of the progressive jackpot is 53.06% plus 2.92% for each $10,000 in the meter and 0.57% for each extra player at the table. The break-even jackpot, when no other players are at the table is $160,530.53.
20 40 Stud Poker Rules
Acknowledgment
My thanks to Stephen How of Discount Gambling for his help with the strategy.