- V1 Is
it possible to gain an advantage at Video Poker?
- V2 What
is the "basic strategy" for Jacks or Better Video Poker?
- V3 What
is the "basic strategy" for Deuces Wild Video Poker?
Section M: Miscellaneous
- M1 How
is Baccarat played?
- M2 How
is Red Dog played?
- M3 How
is Caribbean Stud Poker played?
- M4 Can
the lottery be beat when the jackpot gets high enough?
- M5 How
is Pai Gow Poker played?
- M6 How
is Let It Ride played?
Section V: Video Poker
The video poker strategy discussed here is for the common "8/5" machines
(called 8/5 because of the 8-for-1 payoff for a full house and 5-for-1
payoff for a flush). "Joker's Wild" and "Deuces Wild" machines will
require a much different strategy.
- Q:V1 Is it possible to gain an advantage at
- A:V1 (Steve Jacobs)
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
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
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
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
44.993% 4.317 + royal
Strategy based on the following payoffs:
- Q:V2 What is the "basic strategy" for Jacks
or Better Video Poker?
- A:V2 (Steve Jacobs)
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
0 (2500) royal flush
1 ( 54) 4/royal (break up KQJT9 str-flush) 
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)
 Keep KQJT9 straight flush if progressive jackpot is below 2282 bets.
 Keep two high pair if progressive jackpot is below 2100 bets.
 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
Based upon the following payout schedule:
- Q:V3 What is the "basic strategy" for Deuces
Wild Video Poker?
- A:V3 (Derek Franks)
Royal Flush 800
4 deuces 200
Wild Royal 25
Straight Flush 9
Full House 3
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: Miscellaneous
Baccarat 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.
- Q: How is Baccarat played?
- A:M1 (Steve Jacobs, Steve Brecher)
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
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
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
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
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"
"Red Dog" is also known as "Acey-Deucey" or "between the sheets".
It is a card game that is usually dealt from a shoe containing four
or five decks, although single deck games can be found occasionally,
as can games with 6 or 8 decks.
- Q:M2 How is Red Dog played?
- A:M2 (Steve Jacobs)
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
pair 11:1 (w/ matching 3rd card)
pair push (w/ non-matching 3rd card)
0 (adjacent) push
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.
The player antes, and is then dealt a five-card hand; the dealer is
also dealt five cards of which only one is exposed. The player now
either folds, losing his ante, or bets an additional amount equal
to exactly twice the ante. The dealer then reveals his remaining four
cards. If the dealer's hand is not Ace-King or better, the player
is paid even money on the ante and nothing (i.e., a push) on the bet.
If the dealer's hand is Ace-King or better it is said to "qualify"
(for play against the player). In that case if the dealer's hand is
better than the player's, the player's ante and bet are collected
by the house. If the dealer's qualifying hand is worse than the player's
hand, the player is paid even money on the ante and an amount on the
bet according to the player's hand as follows:
- Q:M3 How is Caribbean Stud Poker played?
- A:M3 (Steve Brecher)
AK or pair 1:1
two pair 2:1
three of a kind 3:1
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
$10,000 0.26 0.71
20,000 0.29 0.74
50,000 0.38 0.82
75,000 0.45 0.90
100,000 0.52 0.97
110,542 0.55 1.00
150,000 0.67 1.12
200,000 0.82 1.26
250,000 0.96 1.41
263,228 1.00 1.45
400,000 1.40 1.85
500,000 1.69 2.14
- If the jackpot payoffs are different, you can calculate the
expectation from the following formula:
- 0.0019654*flush$ + 0.0014406*fullHouse$ + 0.00024010*quads$
+ f(0.00000013852*straightFlush%*JP, straightFlushCap$) + 0.0000015391*JP
- --where * denotes multiplication, JP is the size of the jackpot,
and f(x,y) is equal to the smaller of x and y if there is a cap
on the straight flush payout or equal to x if there is no cap.
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)
(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
(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)
of higher value than dealer's upcard.
(3) AKJ with any card in player's hand matching dealer's upcard
(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.
If "beating" the lotto means having a payback/risk ratio of greater
than 1, I would say that state lottos are definitely beatable.
- Q:M4 Can the lottery be beat when the jackpot
gets high enough?
- A:M4 (David Guercio)
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
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:
- People tend to play birthdays. Don't pick any number less than
- People sometimes will play geometric sequences on the card,
such as rows, and columns, and diagonals. Don't pick these either.
- Even educated people will refuse to play a numeric sequence,
such as 32-33-34-35-36-37, because they think that it isn't "random
enough". Sequences are good to pick, as long as they do not occupy
a single row or column.
Pai-gow poker is a banking poker game played in Las Vegas and some
of the California card clubs. The object of pai-gow poker is to make
two poker hands that beat the banker's hands. The player is dealt
7 cards that he makes into a five card hand (high hand) and a two
card hand (low hand). The hands are played and ranked as traditional
poker hands (with one exception: A2345 is the second highest straight),
and the 5 card hand must be higher than the 2 card hand. If both hands
are better than the banker's hand, you win, if both lose, you lose,
otherwise it's a push. The banker wins absolute ties (i.e. K Q vs
- Q:M5 How is Pai Gow Poker played?
- A:M5 (John F. Reeves)
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,
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
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.
Each player puts up three bets of identical size and is dealt three
cards; two more cards are dealt face-down in front of the dealer.
After examining his three cards, the player may elect to have one
bet returned or to "let it ride." One of the down cards is then turned
over, and then the player may again elect to have one bet returned
-- this election is independent of the prior election. Up to this
point players are not allowed to disclose their three-card hands to
each other. Now the second down card is turned over; the player's
three cards and the two common cards in front of the dealer comprise
a five-card poker hand. The player is paid on each of his one, two
or three remaining bets according to the following schedule:
- Q:M6 How is Let It Ride played?
- A:M6 (Steve Brecher)
Pair of 10s or better 1:1
Two pair 2:1
Three of a kind 3:1
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.