Chances Of Winning Poker At Casino
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Introduction
Four Poker is a new poker variation invented by Roger Snow and marketed by Shufflemaster. The game is similar to Three Card Poker but as the title suggests, four cards are used instead of three. Also, there is no dealer qualifying hand and the player can raise up to three times his ante. However, the dealer gets one extra card to form his best hand.
Video Poker Odds. In order to understand the odds of winning at video poker, players need to become familiar with the game and its rules. Video poker is based on the simple 5-card draw poker, which is one of the simplest forms of table poker. It is played on either a video console, similar to the slot machine or as a virtual game in online casinos. However, thanks to 8 and 6 of spades James Bond has got an open-ended straight flush draw and it means that both 4 and 9 of spades will secure him an invincible poker hand and thus seal his victory in the tournament. On the other hand the winning chances of Le Chiffre drop at this stage. 1st player: 15.00% 2nd player: 47.68%. Blackjack has the best odds of winning, with a house edge of just 1 percent in most casinos, Bean said. Plus, you are playing against only the dealer, not hooded poker champions. 'Blackjack is one. To calculate your poker equity - or how often you should win a hand, you can use a simple formula. Count how many outs you have. For example, if you're drawing to a flush, you have 13 suited cards, two in your hand, two on the board - leaves 9 outs. The chance of you hitting on the turn is 9.4 (+4) = 40%.
Rules
- Two initial bets are available: The Ante and the Aces Up.
- All players get five cards each and the dealer gets six cards. One of the dealer cards is placed face up, and five face down.
- Players making the Ante bet must decide to fold or raise.
- If the player folds he forfeits his Ante bet. He may or may not forfeit his Aces Up bet, depending on casino rules. It shouldn't matter because if the player has a paying Aces Up bet, he shouldn't be folding anyway.
- If player raises, then he must raise at least the amount of the Ante and at most, three times the Ante.
- The player keeps his best four cards and discards one.
- Following is the ranking of hands from lowest to highest: high card, pair, two pair, straight, flush, three of a kind, straight flush, four of a kind.
- After all decisions have been made, the dealer will turn over his cards and select the best four out of six.
- The player's hand shall be compared to the dealer's hand, the higher hand winning.
- If the dealer's hand is higher, then the player shall lose the Ante and Raise.
- If the player's hand is higher or equal then the Ante and Raise shall pay one to one.
- If the player has at least a three of a kind, then he shall also be paid a Bonus, regardless of the value of the dealer's hand. Two different pay tables are available for the Bonus, as displayed below, and are based on the ante bet. Pay Table 1 is the only one I know of to be actually used.
- Another bet is available (similar to the Pairplus in Three Card Poker), based only on the player's four card hand, called the Aces Up. Seven pay tables are available as indicated below. The only one I know of to be actually used is pay table 5.
Bonus Pay Table
| Hand | Table 1 | Table 2 |
|---|---|---|
| Four of a kind | 25 | 30 |
| Straight flush | 20 | 15 |
| Three of a kind | 2 | 2 |
Aces Up Pay Table
| Hand | Table 1 | Table 2 | Table 3 | Table 4 | Table 5 | Table 6 | Table 7 |
|---|---|---|---|---|---|---|---|
| Four of a kind | 50 to 1 | 50 to 1 | 50 to 1 | 50 to 1 | 50 to 1 | 50 to 1 | 50 to 1 |
| Straight flush | 40 to 1 | 40 to 1 | 30 to 1 | 30 to 1 | 40 to 1 | 40 to 1 | 40 to 1 |
| Three of a kind | 9 to 1 | 7 to 1 | 9 to 1 | 7 to 1 | 8 to 1 | 8 to 1 | 7 to 1 |
| Flush | 6 to 1 | 6 to 1 | 6 to 1 | 6 to 1 | 5 to 1 | 6 to 1 | 5 to 1 |
| Straight | 4 to 1 | 5 to 1 | 4 to 1 | 5 to 1 | 4 to 1 | 4 to 1 | 4 to 1 |
| Two pair | 2 to 1 | 2 to 1 | 2 to 1 | 2 to 1 | 3 to 1 | 2 to 1 | 3 to 1 |
| Pair of aces or better | 1 to 1 | 1 to 1 | 1 to 1 | 1 to 1 | 1 to 1 | 1 to 1 | 1 to 1 |
Of these pay tables for the Aces Up side bet, number five is the most popular. The only exceptions that I'm aware of are an unconfirmed report that that Tulalip in Washington uses pay table 4 and the Grand Casino Hinckley in Minnesota uses pay table 1.
Analysis
The following return table is based on optimal player strategy under the 2-20-25 Ante Bonus pay table. The lower right cell shows a house edge of 2.79%.
Return Table Based on Optimal Strategy
| Player Hand | Raise/Fold | Win/Loss | Combinations | Probability | Pays | Return |
|---|---|---|---|---|---|---|
| Four of a Kind | 3 | Win | 40,182,878,736 | 0.000240 | +29 | 0.006960 |
| Four of a Kind | 3 | Lose | 18,594,576 | 0.000000 | +21 | 0.000002 |
| Straight Flush | 3 | Win | 133,224,330,456 | 0.000796 | +24 | 0.019096 |
| Straight Flush | 3 | Lose | 265,177,080 | 0.000002 | +16 | 0.000025 |
| Three of a Kind | 3 | Win | 3,675,379,352,400 | 0.021951 | +6 | 0.131703 |
| Three of a Kind | 3 | Lose | 103,559,138,928 | 0.000618 | -2 | -0.001237 |
| Flush | 3 | Win | 6,599,621,152,728 | 0.039415 | +4 | 0.157660 |
| Flush | 3 | Lose | 784,564,849,080 | 0.004686 | -4 | -0.018743 |
| Straight | 3 | Win | 5,257,469,039,688 | 0.031399 | +4 | 0.125597 |
| Straight | 3 | Lose | 1,301,555,952,216 | 0.007773 | -4 | -0.031093 |
| Two Pair | 3 | Win | 5,539,444,298,496 | 0.033083 | +4 | 0.132333 |
| Two Pair | 3 | Lose | 2,420,447,417,280 | 0.014456 | -4 | -0.057823 |
| One Pair | 3 | Win | 14,764,551,298,548 | 0.088179 | +4 | 0.352714 |
| One Pair | 3 | Lose | 10,806,299,820,804 | 0.064539 | -4 | -0.258155 |
| One Pair | 1 | Win | 13,535,004,289,296 | 0.080835 | +2 | 0.161671 |
| One Pair | 1 | Lose | 22,887,448,286,136 | 0.136691 | -2 | -0.273382 |
| One Pair | Fold | Fold | 5,495,692,732,992 | 0.032822 | -1 | -0.032822 |
| High Card | 1 | Win | 148,058,445,132 | 0.000884 | +2 | 0.001769 |
| High Card | 1 | Lose | 422,493,233,796 | 0.002523 | -2 | -0.005047 |
| High Card | Fold | Fold | 73,523,856,056,112 | 0.439108 | -1 | -0.439108 |
| Totals | 167,439,136,344,480 | 1.000000 | -0.027879 |
The average final bet under optimal strategy is 2.142342 units, making the element of risk, -0.027879/2.142342 = 1.30%. The standard deviation, relative to the original bet, is 2.71.
Beginner Strategy
A simple strategy to this game, first proposed by Stanley Ko, is as follows.
- Raise 3X with a pair of tens or higher.
- Raise 1X with a pair of twos to nines.
- Fold all other.
According to the second edition of 'Beyond Counting' by James Grosjean, this 'simple strategy' results in a house edge of 3.396%.
Intermediate Strategy
The following intermediate strategy was created to balance power and simplicity by our own JB.
- Pair of Aces or better: Bet 3X
- Pair of Js, Qs, Ks: Bet 3X if dealer's upcard is lower than your pair or matches a rank in your hand, otherwise bet 1X
- Pair of 9s, 10s: Bet 1X if dealer's upcard outranks your pair, otherwise bet 3X
- Pair of 8s: Bet 3X if dealer's upcard is a 2, otherwise bet 1X
- Pair of 3s, 4s, 5s, 6s, 7s: Bet 1X
- Pair of 2s or AKQ: Bet 1X if dealer's upcard matches a rank in your hand, otherwise fold
- All other: Fold
Against the 2-20-25 Ante Bonus pay table, the house edge is 2.8526% and the element of risk is 1.3233%.
Advanced Strategy
I'm proud to present the following advanced strategy, also created by my sidekick JB.
- Pair of Aces or better: Bet 3X
- Pair of Ks: Bet 3X, except bet 1X against an Ace and you don't have an Ace nor 4.
- Pair of Js or Qs: Bet 3X, except bet 1X if the dealer's card outranks pair your pair rank and does not match a singleton in your hand.
- Pair of 9s or 10s: Bet 3X, except bet 1X if dealer card outranks your pair rank.
- Pair of 8s: Bet 1X, except bet 3X against a 2
- Pair of 4s thru 7s: Bet 1X
- Pair of 3s: Bet 1X, except fold against a Jack if your highest kicker is a 10 or lower
- Pair of 2s or AKQ: Fold, except bet 1X if dealer card matches a rank in your hand
- AKJT: Fold, except bet 1X against a Jack
- AKJ9 or lower: Fold
Against the 2-20-25 Ante Bonus pay table, the house edge is 2.8498% and the element of risk is 1.3216%. Here is a house edge comparison of various known strategies.
- Simple: 3.396%
- Intermediate: 2.853%
- Advanced: 2.850%
- Optimal: 2.788%
To put it another way, here are the cost of errors:
- Simple: 0.606%
- Intermediate: 0.065%
- Advanced: 0.062%
- Optimal: 0.000%
Aces Up Analysis
The next table shows the probability of each hand and the return under pay table five of the Aces Up side bet. The lower right cell shows a house edge of 3.89%.
Return for Aces Up Pay Table 5
| Hand | Combinations | Probability | Pays | Return |
|---|---|---|---|---|
| Four of a kind | 624 | 0.00024 | 50 | 0.012005 |
| Straight flush | 2072 | 0.000797 | 40 | 0.03189 |
| Three of a kind | 58656 | 0.022569 | 8 | 0.180552 |
| Flush | 114616 | 0.044101 | 5 | 0.220504 |
| Straight | 101808 | 0.039173 | 4 | 0.15669 |
| Two pair | 123552 | 0.047539 | 3 | 0.142617 |
| Pair of aces | 81096 | 0.031203 | 1 | 0.031203 |
| Nothing | 2116536 | 0.814378 | -1 | -0.814378 |
| Total | 2598960 | 1 | -0.038917 |
The next table shows the house edge according to all four Aces Up pay tables.
Winning At The Casino
Aces Up House Edge
| Pay Table | House Edge |
|---|---|
| 1 | 1.98% |
| 2 | 2.58% |
| 3 | 2.78% |
| 4 | 3.37% |
| 5 | 3.89% |
| 6 | 4.24% |
| 7 | 6.15% |
Note: There is also a similar game called Crazy Four Poker.
Acknowledgments
Chances Of Winning At Slots
I would like to recognize:
Odds Of Winning Poker At A Casino
- JB for the analysis of the optimal strategy.
- Stanley Ko for the simplified strategy.
- James Grosjean for the unpublished advanced strategy.