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Section V: Video Poker
In order to have an advantage over the house, you must find a machine with a progressive jackpot that is larger than about 1750 maximum bets. ($8750 for $1 machines, $2200 for $.25 machines, $440 for $.05 machines). This level only makes the game even with the house. The jackpot must be higher than this in order to gain an advantage. The player's edge increases by about 1% for each addition of 350 maximum bets into the progressive jackpot. In order to have a 2% edge, the jackpot must be about 2500 max. bets. ($12,500 for $1 machines, $3125 for $.25 machines, $625 for $.05 machines). The main difficulty with playing video poker is that it takes an average of 60 hours of rapid play to hit a royal flush, and it takes a _huge_ bankroll to survive long enough to win. During this time, the casino enjoys an advantage of approximately 5%. Straight flushes can be expected about once every 6 hours on average, but these contribute only about 0.5% to the player's return. 4-of-kind hands occur only about once per hour, and these hands account for about 5% of the player's return. What this all means to the video poker player is that you will be playing with about a 10% disadvantage while waiting for an occasional "boost" from a 4-of-kind or straight flush. On average, it will take a bankroll about as large as the progressive jackpot to survive long enough to hit the royal flush (and this assumes that the jackpot is large enough to give the player a reasonable edge over the house). The following table shows the relative frequency of each hand, and the resultant effect on the expected return, assuming the given strategy is used. The table shows that you can expect to get nothing back about 55% of the time, and hit either a high pair, two pair, or three of a kind another 41% of the time. Hands of higher value occur only about 3.6% of the time. This means that the house has a whopping 31% edge most of the time. return % rate frequency variance
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5.308 -> 0.00306 -> 1/32680 91.90 --=<ROYAL FLUSH!!!>=--
0.492 -> 0.00984 -> 1/10163 0.246 STRAIGHT FLUSH!!!!
5.878 -> 0.235 -> 1/425 1.469 FOUR OF A KIND!!!
9.183 -> 1.148 -> 1/87 0.735 FULL HOUSE!!
5.584 -> 1.117 -> 1/89.5 0.293 FLUSH!
4.512 -> 1.128 -> 1/88.7 0.180 STRAIGHT!
22.227 -> 7.409 -> 1/13.5 0.667 THREE OF A KIND
25.780 -> 12.890 -> 1/7.76 0.516 TWO PAIR
21.053 -> 21.053 -> 1/4.75 0.211 HIGH PAIR
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44.993% 4.317 + royal
high pair 1 for 1
two pair 2 for 1
3 kind 3 for 1
straight 4 for 1
flush 5 for 1
full house 8 for 1
4 kind 25 for 1
str flush 50 for 1
royal flush 2500 for 1 (expected return 102%)
Simplified strategy (find first hand that matches, keep only needed cards). Best draws are listed in order of decreasing expected value. Expected value of each draw is shown, in units of one max. bet. Numbers in () vary, depending on progressive jackpot (value shown is for jackpot of 2500 max. bets). drawing value hand
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0 (2500) royal flush
1 ( 54) 4/royal (break up KQJT9 str-flush) [1]
0 50 straight flush
0 25 4 kind
0 8 full house
0 5 flush
2 4.24 3 kind
0 4 straight
1 3.4 4/str-flush
2 ( 2.9) 3/royal (break up pairs) [2,3]
1 2.51 two pair
3 1.53 high pair
1 1.0 4/flush
1 0.87 KQJT 4/straight
3 0.814 low pair
1 0.809 QJT9 4/straight (outside, two high cards)
1 0.745 JT98 4/straight (outside, one high card)
2 0.699 QJ9 3/str-flush
2 0.697 JT9 3/str-flush
3 ( 0.69) 2/royal (both non-tens)
1 0.681 4/straight (outside, no high cards)
2 0.599 3/str-flush (one high card, spread 4)
2 0.597 3/str-flush (spread 3)
3 ( 0.59) 2/royal (10 + one high card)
1 0.596 AKQJ straight (4 high cards)
1 0.532 AKQT/AKJT/AQJT/KQJ9 straight (3 high cards)
2 0.515 KQJ unsuited
3 0.509 QJ unsuited
2 0.502 3/str-flush (one high card, spread 5)
2 0.500 3/str-flush (none high cards, spread 4)
3 0.48 3 unsuited high cards (keep lowest two)
3 0.48 2 unsuited high cards
4 ( 0.48) high card
2 0.402 3/str-flush (none high cards, spread 5)
5 0.360 garbage (draw 5 new cards)
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[1] Keep KQJT9 straight flush if progressive jackpot is below 2282 bets.
[2] Keep two high pair if progressive jackpot is below 2100 bets.
[3] Keep high pair plus paired 10's if progressive is below 2175 bets.
The following draws should NOT be taken, since drawing 5 new
cards gives a greater expected gain.
1 0.340 4/straight (inside, no high cards) --> keep none
2 0.305 3/flush (no high cards) --> keep none
2 0.275 3/straight (no high cards) --> keep none
Royal Flush 800
4 deuces 200
Wild Royal 25
5-of-a-kind 15
Straight Flush 9
4-of-a-kind 5
Full House 3
Flush/Straight 2
3-of-a-kind 1
Average payback is 100.761% The following strategy yields an average profit of 350 units per average royal cycle of 45,278 hands. #d Hand Type Expected Value 4 Four deuces 200 3 Royal Flush(wild) 25 3 5-of-a-Kind(10-A)* 15 3 deuces alone 15.026 2 Royal Flush(wild) 25 2 5-of-a-Kind 15 2 Straight Flush 9 2 4-of-a-Kind 5.851 2 Royal Flush 4 4.606 2 Straight Flush 4 3.340 2 deuces alone 3.260 1 Royal Flush 25 1 Straight Flush 15 1 4-of-a-Kind 5.851 1 Royal Flush 4 3.501 1 Full House 3 1 Straight Flush 4 2.209 1 3-of-a-Kind 2.018 1 Flush or Straight 2 1 Straight Flush 4 i 1.974 1 Straight Flush 4 di 1.698 1 Straight Flush 4i ace 1.421 1 Royal Flush 3 1.098 1 Straight Flush 3 1.091 1 deuce alone 1.029 0 Royal Flush 800 0 Royal Flush 4 19.626 0 Straight Flush 9 0 4-of-a-Kind 5.851 0 Full House 3 0 3-of-a-Kind 2.018 0 Flush or Straight 2 0 Straight Flush 4 1.643 0 Straight Flush 4i 1.370 0 Royal Flush 3 1.325 0 Straight Flush 4i ace 1.106 0 one pair ** .561 0 Straight Flush 3 .520 0 Flush 4 or Straight 4 .511 0 Straight Flush 3 i .438 0 J-10 suited .362 0 Straight Flush 3 di .355 0 Straight 4 i .340 0 Q-J or Q-10 suited .332 0 garbage - draw 5 .322 * Don't break up 5-of-a-kinds of tens through aces. The removal of those 2 cards reduces the wild royal possibilities. OTOH, discarding two low cards makes 3 deuces alone worth 15.06. ** Never draw to 2 pair. Discard either pair and draw 3. Section M: MiscellaneousBaccarat is a card game that is dealt from a shoe that holds 6 or 8 decks of cards. Two hands are dealt by the house dealer, the "banker" hand and the "player" hand. Before the hands are dealt, bets may be placed on the banker hand, on the player hand, or on a tie. Winning bets on banker or player are paid 1:1, but a commission of 5% is charged on bank bets making the net odds on such bets 0.95 to 1. Some casinos may charge a lower commission (e.g., at this writing, Binion's Horseshoe in Las Vegas charges 4%.). Some sources report that tie bets are paid 8:1, while others claim that tie bets are paid 9:1, so this may vary from casino to casino. If there is a tie, bets on the banker or player are returned. Once a bet has been placed, there are no opportunities for further decisions -- both the banker hand and the player hand are dealt according to fixed rules, resulting in final hands of either two or three cards for each. The value of a hand is determined by adding the values of its individual cards. Tens and face cards are counted as zero, while all other cards are counted by the number of "pips" on the card face. Only the last digit of the total is used, so all baccarat hands have values in the range 0 to 9 inclusive. The hand with the higher value wins; if the hands have the same value, the result is a tie. A game is started by dealing two cards for the player hand and two cards for the bank hand. An initial hand with a value of 8 or 9 is called a "natural." If either hand is a natural, its holder must expose it and the game ends. Otherwise play continues, first with the player hand and then with the banker hand, according to the following rules. Rules for the player hand: If the player's first two cards total 6 or more, then the player must stand without drawing a card. If the player's first two cards total 5 or less, the player must draw one additional card. Rules for the banker hand: If the banker's first two cards total 7 or more, then the banker must stand without drawing a card. If the banker's first two cards total 0, 1, or 2, then the banker must draw one card. If the banker's first two cards total 3, 4, 5, or 6, then whether the banker draws is determined by the whether the player drew, and if so the value of the player's draw card, as shown by the table below. Bank Drawing vs. player's draw
Bank N 0 1 2 3 4 5 6 7 8 9 <--- player's draw card
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9 - - - - - - - - - - -
8 - - - - - - - - - - -
7 - - - - - - - - - - -
6 - - - - - - - D D - -
5 D - - - - D D D D - -
4 D - - D D D D D D - -
3 D D D D D D D D D - D
2 D D D D D D D D D D D
1 D D D D D D D D D D D
0 D D D D D D D D D D D
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D = draw, N = no card drawn by player
The probability distribution for a hand dealt from a complete shoe is as follows: Probability Probability of Probability
of bank win of player win of tie
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6 decks 0.458652719 0.446278570 0.095068711
8 decks 0.458597423 0.446246609 0.095155968
This implies the following house advantages: Bet bank Bet bank Bet player Bet tie Bet tie
decks 5% vig. 4% vig. 9:1 8:1
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6 1.05585% 0.59720% 1.23741% 4.93129% 14.43816%
8 1.05791% 0.59931% 1.23508% 4.84403% 14.35963%
Edward O. Thorp and others have determined that card counting is not effective in overcoming the house edge at the baccarat tables. Compared to blackjack, card counting is about 9 times less effective when used against baccarat. See Thorp's "The Mathematics of Gambling" for details.
After the players bet, two cards are dealt face up on the table. If the two cards are adjacent, it is a tie. If the two cards are not identical, the player is allowed to place a "raise" bet, up to the size of the original bet. If the third card drawn is _between_ the first two cards, the player wins. If the first two cards are identical the player is not allowed to raise, and if the third card matches the first two, the player is paid 11:1. Payoffs are at even money unless the first two cards are a pair or the "spread" is 3 or less.
Spread Payoff
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pair 11:1 (w/ matching 3rd card)
pair push (w/ non-matching 3rd card)
0 (adjacent) push
1 5:1
2 4:1
3 2:1
4 - 11 1:1
The number of players at the table is totally irrelevant, since all players win or lose simultaneously. The only strategy decision that the player is allowed to make is whether or not to double the bet. With these payoffs, the bet should be doubled only when the spread is 7 or greater. The house edge for Red Dog is about 3%, and decreases slightly as more decks are used.
AK or pair 1:1
two pair 2:1
three of a kind 3:1
straight 4:1
flush 5:1
full house 7:1
four of a kind 20:1
straight flush 50:1
royal flush 100:1
There is an optional independent side bet of $1.00 available for which the player is paid for being dealt premium hands (flush or better); the payoff of this side bet is based on a progressive jackpot for straight flushes (10% of jackpot) and royal flushes (100%), although some places cap the straight flush payoff (e.g., $5000 max). The jackpot bet is extremely unfavorable except for the case of a very large jackpot. If the jackpot payoff is $50/75/100 for flush/full house/quads and there is no straight flush cap, then the expected return per $1 jackpot bet is approximately $0.23 plus 2.924 cents for each $10,000 in the jackpot; if the flush/ full house/quads payoff is $100/250/500, the expected return is approximately $0.68 plus 2.924 cents for each $10,000 in the jackpot. Examples: Jackpot Expectation per $1 bet
------- 50/75/100 100/250/500 --flush/full/quads payoffs
--------- -----------
$10,000 0.26 0.71
20,000 0.29 0.74
50,000 0.38 0.82
75,000 0.45 0.90
100,000 0.52 0.97
110,542 0.55 1.00
150,000 0.67 1.12
200,000 0.82 1.26
250,000 0.96 1.41
263,228 1.00 1.45
400,000 1.40 1.85
500,000 1.69 2.14
My analysis of the basic game: When the dealer doesn't qualify the player's bet wins the ante and the dealer's payoff on the ante. In other words, if the dealer doesn't qualify the player is paid even money on the bet. However, in the long run the dealer will qualify 56.3% of the time. A bluff is always an unfavorable bet. Even the best possible bluff--where the player holds an Ace or King, another card which matches the dealer's upcard, and a four-flush of the same suit as the dealer's upcard--is unfavorable. This means that a player who always folds hands worse than Ace-King will lose less in the long run than one who sometimes bluffs. A pair or better should always be bet. A bet on even the worst possible pair--deuces, with no Ace nor King, no card matching the dealer's upcard, and no card of the same suit as the dealer's upcard--yields an expected profit. This means that a player who always bets a pair of deuces or better will lose less in the long run than one who sometimes folds such hands. The dealer will fail to qualify 43.7% of the time, and will qualify with an Ace-King (no pair) 6.4% of the time. The player who holds an Ace-King and bets will win even money more than 43.7% of the time (because the player's holding Ace-King reduces the chance of the dealer qualifying), and will be paid two to one (1:1 bet payoff plus 0.5:1 ante plus 0.5:1 ante payoff) when the player's Ace-King beats the dealer's. Therefore, there are some player Ace-King hands which should be bet, depending on what other cards the player holds. For example, if the player holds a card having the same value as the dealer's upcard, the chance of the dealer having a pair is reduced. The optimum strategy is to bet when the player holds: (1) AKQJ or better (including any pair or better)
or
(2a) AKQxx with any card in player's hand matching dealer's upcard; or
(2b) with both x cards having higher value than dealer's
upcard; or
(2c) with a four flush of the same suit as dealer's upcard and:
at least one of the x cards being either:
8 or better (i.e., 8, 9, or 10)
or
of higher value than dealer's upcard.
or
(3) AKJ with any card in player's hand matching dealer's upcard
or
(4) AKxxx with any x card matching dealer's upcard
The results of this strategy and two simpler strategies are shown below, each based on computer simulation of 200 million deals. "Expected loss per ante amount per hand" is the average amount that the player will lose per hand in the long run as a percentage of the ante amount. "Payback per $1 risked" is the average long run total payback on each dollar wagered--on antes plus bets. Expected loss per Strategy Bet frequency ante amount per hand Payback per $1 risked Optimum 52.0% 5.23% $0.9743 Bet any pair or better 49.9% 5.48% $0.9726 Bet Ace-King or better 56.3% 5.75% $0.9729 For the casual player, "Bet any pair or better" is the recommended strategy. The expected difference in total loss versus the optimum strategy over a couple of hundred hands is about half of one ante. "Bet Ace-King or better" provides more betting action at the cost of another half an ante per couple of hundred hands.
In Texas Pick-6 lotto, you pick 6 mutually exclusive numbers from 1 to 50. That gives you approximately 1/16,000,000 chance of winning. Many people do not play until the lotto jackpot goes over $16,000,000, as a result. It's a little more complicated than that though, because the money is paid out over 20 years, and you have to account for inflation. The actual value of the money you get paid is (assuming constant %5 inflation) is the jackpot divided by 20 times the sum from 0 to 19 of (.95)**N, where N is the summation index. The sum is 12.83, in this example, so you really need to wait until the lotto is (20/12.83)*16,000,000, or approximately $25 million. Texas Pick-6 frequently exceeds this total, but resets to $3 million when somebody wins. Of course, all this is predicated on being the sole winner of a $25 million lotto, or at least, say, winning $75 million and splitting with at most two other people. You can reduce the number of people that you split with by picking the numbers that nobody else does. I use this formula in picking numbers:
The game is played with a 52 cards plus one joker. The joker can be used as an Ace or to complete a flush or straight. The table layout has 7 spots one in front of the dealer and 6 for players, like this: Dealer
7
1 6
2 5
3 4
Each player spot has spaces for a bet, low hand, high hand and sometimes the house commission. The dealer deals 7 7-card hands in front of the chip tray. The banker can be a player, but is usually the house. The banker designates which hands go to which player by shaking a dice cup with three dice; the banker's position is either 1, 8 or 15 and the hands are passed out counterclockwise. So, if the dealer is the bank and the dice total to 6, player 5 gets the first hand, player 6 gets the second, the dealer gets the third and so on. The dice mumbo-jumbo appears to be ritual stuff --- you don't need to worry about anything until you get your hand. The player puts the two card hand face down in the box closest to the dealer, and the five card hand face down in back. Once everybody has set their hand, the dealer turns over and sets the bank's hand. The dealer goes counterclockwise around the table comparing the banks hand to the players, and taking, paying, or knocking. There is a 5% commission on winning bets that you can either put out next to your winning bet, or the dealer will subtract from your payoff. The lowest minimum bet is $5, seen at the Imperial Place and Four Queens. In pai-gow poker, the only strategic decisions are how much to bet and how to set your hand. The simple basic strategy for setting your hand is to make the highest 2-card hand that is less than your five card hand. If you can't figure out what to do, you can show your hand to the dealer and they will tell you how the house would set it. Since pairs generally win the 2-card hands, and two-pair wins the 5-card hands, the only difficult decisions are when to split two pairs. The house rules at the Four Queens were not to split low pairs (<= 6) and not to split pairs <= 10 if there was a Ace high two card hand. So the house would set A 10 10 6 6 5 3 => A 5 / 10 10 6 6 3
K Q 10 10 6 6 3 => 6 6 / 10 10 K Q 3
A ``Pai-gow'' is a hand with no pairs, such as Q J / K 7 8 6 2. Things get a little weird if a player wants to be the bank. To quote from the IP house rules: ``The House Dealer or the player may be the ``BANKER.'' The Bank wagers against all players. The bank will alternate between the house and the player (the House Dealer will at least take the bank every other hand). The BANKER will be signified by a white plastic marker. A Bank Player must either cover half or all wagers against him/her. The House will co-bank at 50/50 only at the Bank Player's request. The hand will be set according to house way and the table limit will apply if the House acts as a co-banker. In order to bank, a player must have played the previous hand against the House. The House will wager a sum equal to that player's wager against the house the previous hand. The player may request that a smaller amount be wagered. A Banker must be bank at the same spot of the hand he previously played against the house.'' Got that?? In the CA card clubs, all wagering is between players, so the option to be the bank rotates among the active players. The rule differences from the IP rules are that the Joker is wild, and the house commission is a flat $1 per hand ($10 minimum bet). Pai-gow poker is an easy game to play, and since each hand takes a while to play (dealer has to shuffle for each game) and most hands push, you can play on $20 at a $5 table for quite a while.
Pair of 10s or better 1:1
Two pair 2:1
Three of a kind 3:1
Straight 5:1
Flush 8:1
Full House 11:1
Four of a Kind 50:1
Straight Flush 200:1
Royal Flush 1000:1
Being able to have up to two of the three bets returned by the dealer is logically equivalent to starting with one bet and being allowed to put out up to two more. I surmise that the game is structured as it is because it would otherwise be too easy for players to covertly press bets -- the bet circles on the layout are quite close together. The optimal strategy for this game is as follows. On the first three cards, take back a bet unless one holds: --a pair of 10s or better, or three of a kind; or
--three cards to a straight flush, provided:
--contiguous and 543 or higher, or
--one "hole" and at least one card is 10 or higher, or
--two "holes" and at least two cards are 10 or higher.
On the fourth card, take back a bet unless one has: --a pair of 10s or better, two pair, or three or four of a kind; or
--a four-flush; or
--an open-ended straight including a 10 or higher.
The following bets are optional, i.e., expected return = 1.000...
--an open-ended straight not including a 10 or higher; or
--all cards 10 or higher (an inside A-to-10 straight).
Playing this strategy provides an expected return of 0.971352 per unit bet. The average bet per hand is 1.223707 units (where one to three units are bet per hand and no optional bets are made), and the average unit cost per hand is 0.035057.
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legal disclaimer