Some forms of poker, often called lowball, sometimes called “low poker,” reward poor poker hands (in the traditional sense). There are three common variations on this idea, differing in whether aces are treated as high cards or low cards, and whether or not straights and flushes are used. The methods are:
Ace-to-five low: The lowest possible hand is -4-3-2-A, called a wheel. Aces are low and straights and flushes are ignored. This is the most common method.
Ace-to-six low: Also called 6-4 low, since the lowest possible hand is
-4-3-2-A. Aces are low and straights and flushes count as high hands.
Deuce-to-six low: The other, mostly unused, possibility would be 6-5 low. Aces are high, straights and flushes are ignored.
Some games are played high-low split, where the player with the best traditional poker hand (called the “high hand”) splits the pot with the best low hand. The low hand is decided by one of the methods above. According to Official Rules of Card Games by Albert Morehead[1], the low hand in high-low is generally the deuce-to-seven low, although many on-line casinos use ace-to-five low, with a qualifier, e.g., no card higher than an 8. Low hands tie more frequently than high hands, especially in community card games, so it is not uncommon for such a hand to win a small fraction of a poker pot. For example, if one player has the high hand on showdown, and two other players tie for the best low hand, the high hand wins half of the pot and each low hand wins only a quarter of the pot. Playing ace-to-five high-low greatly increases the chances of the “scoop”–winning both hands–because a low flush or straight may count for both high and low.
References
^ Albert H. Morehead (1996). Official Rules of Card Games. Ballantine Books. ISBN 0449911586.
Many lotteries have a “powerball” (or “bonus ball”). If the powerball is drawn from a different pool of numbers from the main lottery, then simply multiply the odds by the number of powerballs. For example, in the 6 from 49 lottery we have been discussing in this article, if there were 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in 56.66 x 10, or 566.6 (the probability would, of course, be divided by 10, to give an exact value of 8815/4994220).
Where more than 1 powerball is drawn from a separate pool of balls to the main lottery (e.g. the Euromillions game), the odds of the different possible powerball matching scores should be calculated using the method shown in the “other scores” section above (in other words, treat the powerballs like a mini-lottery in their own right), and then multiplied by the odds of achieving the required main-lottery score.
If the powerball is drawn from the same pool of numbers as the main lottery, then, for a given target score, one must calculate the number of winning combinations which includes the powerball. For games based on the Canadian lottery (e.g. Lotto, the UK lottery), after the 6 main balls are drawn, an extra ball is drawn from the same pool of balls, and this becomes the powerball (or “bonus ball”), and there is an extra prize for matching 5 balls + the bonus ball. As described in the “other scores” section above, the number of ways one can obtain a score of 5 from a single ticket is c(6,5)*c(43,1), or 258. Since the number of remaining balls is 43, and your ticket has 1 unmatched number remaining, 1/43 of these 258 combinations will match the next ball drawn (the powerball) – so there are 258/43 = 6 ways of achieving it. Therefore, the odds of getting a score of 5 + powerball are c(49,6)/6 = 1 in 2,330,636.
Of the 258 combinations that match 5 of the main 6 balls, in 42/43 of them the remaining number will not match the powerball, giving odds of c(49,6)/(258*(42/43)) = 166474/3 (approx 55491.33) for obtaining a score of 5 without matching the powerball.
Using the same principle, to calculate the odds of getting a score of 2 + powerball, calculate the number of ways to get a score of 2 as c(6,2)*c(43,4) = 1,851,150 then multiply this by the probability of one of the remaining four numbers matching the bonus ball – which is 4/43. 1,851,150*(4/43) = 172,200, so the probability of obtaining the score of 2 + bonus ball is 172,200/c(49,6) = 1025/83237. This gives approximate decimal odds of 81.2.
A card that isn’t wild or otherwise modified by the game rules. In some houses, a natural hand beats an equivalent hand that uses wild cards, though this is not generally the case.
The fundamental bet in craps is the pass line bet, in which one bets that the dice will pass (that is, roll the point number before rolling a 7). A pass line bet is also won if the come-out roll is a 7 or 11. The following discussion assumes that the shooter, as is usually the case, is betting on the pass line.
If a 4,5,6,8,9, or 10 is thrown on the come-out roll (i.e., if a point is set), most casinos allow pass line bettors to take odds by placing from one to five times (and at some casinos, up to 100 times) the pass line bet behind the line. This additional bet wins if the point is rolled again before a 7 is rolled (the point is made) and pays at the true odds, 2-to-1 if 4 or 10 is the point, 3-to-2 if 5 or 9 is the point, and 6-to-5 if 6 or 8 is the point. While the house has a small (1.4%) advantage on pass line bets, the house has no advantage at all on odds bets. Therefore, taking the maximum odds (which vary by casino) can lower the house percentage for any given bet down to as low as 0.5%.
Odds bets in craps are one of the few bets offered at a casino that are completely free of any house advantage. Another such bet is the “double-up” option offered to the player in some forms of video poker after winning a hand.
Let’s see why that is. There are 36 possible permutations (ways to roll a pair of 6-sided dice):
1-1 = 1 way to make a 2
1-2 2-1 = 2 ways to make a 3
1-3 2-2 3-1 = 3 ways to make a 4, true odds pays 2-1
1-4 2-3 3-2 4-1 = 4 ways to make a 5, true odds pays 3-2
1-5 2-4 3-3 4-2 5-1 = 5 ways to make a 6, true odds pays 6-5
1-6 2-5 3-4 4-3 5-2 6-1 = 6 ways to make a 7
2-6 3-5 4-4 5-3 6-2 = 5 ways to make an 8, true odds pays 6-5
3-6 4-5 5-4 6-3 = 4 ways to make a 9, true odds pays 3-2
4-6 5-5 6-4 = 3 ways to make a 10, true odds pays 2-1
5-6 6-5 = 2 ways to make an 11
6-6 = 1 way to make a 12
There are a total of 36 possible combinations. So on the come-out roll there are 8 ways to win, 4 ways to lose and (36-12=) 24 ways to start a point.
The odds of making the point are the ratio of the number of ways to make a 7 to the number of ways to make the point. For example, there are five ways to make a 6 or 8, so the odds of making a point of 6 or 8 are 6-5. Therefore an odds bet of $5 on 6 or 8 pays out $6.
Many experienced craps players only make pass line and odds bets since the odds are much more favorable to the player than any other bets in craps, and in fact most casino games. The pass line and odds bet is one of the best wagers you can make in gambling, and a lot of the excitement and fortunes of craps can be experienced with this simple bet.
The rules for the come wagers are the same as for the pass line except that they can only be made after the come-out roll. Effectively, they represent starting a new game using the same stream of numbers being generated by the existing (pass line) game.
Because of the come bet, if the shooter makes their point, a player can find themselves in the situation where they have a come bet with odds on it, and yet be rooting for the shooter to roll a 7 on their next come-out roll. Because of this, it is usual that odds bets on come wagers are presumed to be not working. That means that if the shooter rolls a 7 on the come-out roll, any players with active come bets lose their initial wager but will have their odds money returned to them, unless they tell the dealer that they want their odds working. Conversely, if the shooter rolls a number that matches an active come bet, the original bet is paid off at even money and the odds money is returned to the player (unless they told the dealer that they wanted their odds working, in which case they are paid at the true odds).
There is also a don’t come box in which one can place bets that the dice will not pass on the next sequence starting with the immediate roll as a virtual come-out roll; even the shooter may bet that he or she will miss out. Don’t pass and don’t come bets are basically the opposite of pass and come bets; the player is betting that a 7 will be rolled before the point. On the come-out roll a 7 or an 11 is a loss, whereas a 2, 3, or 12 is a win. Casino craps layouts bar either 2 or 12 on the don’t pass and don’t come bets. This means that if 2 is barred and the shooter rolls a 2 on the come-out roll, the wager is a stand off and the player’s money is returned.
When betting against the shooter, the bettor must put up the long side of the bet. Thus a don’t pass bettor who bets $10 when the point is a 4 could place an odds bet of $20 behind the line. If the shooter rolls a 7 before achieving their point, the bettor would receive $10 for the don’t pass bet plus $10 for their odds bet. Even though the house advantage on the don’t pass and don’t come bets are almost identical to the pass line and come bets, for most players the disadvantage of putting up the long side of the bet makes the don’t pass line less desirable. Additionally, many craps players consider don’t pass and don’t come bets to be in poor taste, or even “taboo”.
When evaluating a hand before the flop, it’s useful to have some idea of how likely the hand is dominated. A dominated hand is a hand that is beaten by another hand (the dominant hand) and is extremely unlikely to win against it. Often the dominated hand has only a single card rank that can improve the dominated hand to beat the dominant hand (not counting straights and flushes.) For example, KJ is dominated by KQ—both hands share the king and the queen kicker is beating the jack kicker. Barring a straight or flush, the KJ will need a jack on the board to improve against the KQ (and will still be losing if a queen comes on the board also.) A pocket pair is dominated by a pocket pair of higher rank.
Pocket pairs
Barring a miracle straight or flush, a pocket pair needs to make three of a kind to beat a higher pocket pair.
To calculate the probability that another player has a higher pocket pair, first consider the case against a single opponent. The probability that a single opponent has a higher pair can be stated as the probability that the first card dealt to the opponent is a higher rank than the pocket pair and the second card is the same rank as the first. Where r is the rank of the pocket pair (assigning values from 2–10 and J–A = 11–14), there are (14 − r) × 4 cards of higher rank. Subtracting the two cards for the pocket pair leaves 50 cards in the deck. After the first card is dealt to the player there are 49 cards left, 3 of which are the same rank as the first. So the probability of a single opponent being dealt a higher pocket pair is
P = (((14-r)x4)/50) x (3/49)
The following approach extends this equation to calculate the probability that one or more other players has a higher pocket pair.
Multiply the base probability for a single player for a given rank of pocket pairs by the number of opponents in the hand;
Subtract the adjusted probability that more than one opponent has a higher pocket pair. (This is necessary because this probability effectively gets added to the calculation multiple times when multiplying the single player result.)
Where n is the number of other players still in the hand and Pma is the adjusted probability that multiple opponents have higher pocket pairs, then the probability that at least one of them has a higher pocket pair is
P = ((84-6r)/1225) x n – Pma.
The calculation for Pma depends on the rank of the player’s pocket pair, but can be generalized as
Pma = P
+ 2P
+ … + (n-1)Pn,
where P
is the probability that exactly two players have a higher pair, P
is the probability that exactly three players have a higher pair, etc. As a practical matter, even with pocket 2s against 9 opponents, P
< 0.0015 and P
< 0.00009, so just calculating P
and P
gives an adequately precise result.
The following table shows the probability that before the flop another player has a larger pocket pair when there are one to nine other players in the hand.
Probability of facing a
larger pair when holding
Against 1
Against 2
Against 3
Against 4
Against 5
Against 6
Against 7
Against 8
Against 9
KK
0.0049
0.0098
0.0147
0.0196
0.0244
0.0293
0.0342
0.0391
0.0439
QQ
0.0098
0.0195
0.0292
0.0388
0.0484
0.0579
0.0673
0.0766
0.0859
JJ
0.0147
0.0292
0.0436
0.0577
0.0717
0.0856
0.0992
0.1127
0.1259
TT
0.0196
0.0389
0.0578
0.0764
0.0946
0.1124
0.1299
0.1470
0.1637
99
0.0245
0.0484
0.0718
0.0946
0.1168
0.1384
0.1593
0.1795
0.1990
88
0.0294
0.0580
0.0857
0.1125
0.1384
0.1634
0.1873
0.2101
0.2318
77
0.0343
0.0674
0.0994
0.1301
0.1595
0.1874
0.2138
0.2387
0.2619
66
0.0392
0.0769
0.1130
0.1473
0.1799
0.2104
0.2389
0.2651
0.2890
55
0.0441
0.0862
0.1263
0.1642
0.1996
0.2324
0.2623
0.2892
0.3129
44
0.0490
0.0956
0.1395
0.1806
0.2186
0.2532
0.2841
0.3109
0.3334
33
0.0539
0.1048
0.1526
0.1967
0.2370
0.2729
0.3040
0.3300
0.3503
22
0.0588
0.1141
0.1654
0.2124
0.2546
0.2914
0.3222
0.3464
0.3633
The following table gives the probability that a hand is facing two or more larger pairs before the flop. From the previous equations, the probability Pm is computed as
Pm = P
+ P
+ … + Pn.
Probability of facing multiple
larger pairs when holding
Against 2
Against 3
Against 4
Against 5
Against 6
Against 7
Against 8
Against 9
KK
< 0.00001
0.00001
0.00003
0.00004
0.00007
0.00009
0.00012
0.00016
QQ
0.00006
0.00018
0.00037
0.00061
0.00091
0.00128
0.00171
0.00220
JJ
0.00017
0.00051
0.00102
0.00171
0.00257
0.00360
0.00482
0.00621
TT
0.00033
0.00099
0.00200
0.00335
0.00504
0.00709
0.00950
0.01226
99
0.00054
0.00164
0.00330
0.00553
0.00836
0.01177
0.01580
0.02045
88
0.00081
0.00244
0.00493
0.00828
0.01253
0.01769
0.02378
0.03084
77
0.00112
0.00341
0.00689
0.01160
0.01758
0.02487
0.03351
0.04353
66
0.00149
0.00454
0.00918
0.01550
0.02353
0.03335
0.04503
0.05861
55
0.00191
0.00583
0.01182
0.01998
0.03040
0.04318
0.05840
0.07619
44
0.00239
0.00728
0.01480
0.02506
0.03821
0.05438
0.07371
0.09635
33
0.00291
0.00890
0.01812
0.03075
0.04698
0.06699
0.09099
0.11919
22
0.00349
0.01068
0.02180
0.03706
0.05673
0.08107
0.11034
0.14484
From a practical perspective, however, the odds of out drawing a single pocket pair or multiple pocket pairs are not much different. In both cases the large majority of winning hands require one of the remaining two cards needed to make three of a kind.
Hands with one ace
When holding a single ace (referred to as Ax), it is useful to know how likely it is that another player has a better ace—an ace with a higher second card. The weaker ace is dominated by the better ace. The probability that a single opponent has a better ace is the probability that they have either AA or Ax where x is a rank other than ace that is higher than the player’s second card. When holding Ax, the probability that the other player has AA is (3/50) x (2/49) ~ 0.00245. Where x is the rank 2–K of the second card (assigning values from 2–10 and J–K = 11–13) the probability that a single opponent has a better ace is calculated by the formula
P = ((3/50) x (2/49)) + ((3/50) x (((13-x) x 4)/49) x 2) = (3/1225) + (12 x (13 – x))/1225 = (159 – 12x)/1225.
The probability (3/50) x (((13-x) x 4)/49) of a player having Ay, where y is a rank such that x < y <= K, is multiplied by the two ways to order the cards A and y in the hand.
When facing two opponents, for any given starting hand the number of possible combinations of hands the opponents can have is
(50/2)(48/2) = 1,381800
hands. For calculating probabilities we can ignore the distinction between the two opponents holding A♠J♥ and
♥
♣ and the opponents holding ♥
♣ and A♠J♥. The number of ways that hands can be distributed between n opponents is n! (pronounced n factorial). So the number of unique hand combinations H against two opponents is
H = (50/2)(48/2) -: 2! = 690,900
and against three opponents is
H = (50/2)(48/2)(46/2) -: 3! = 238,360,500
and against n opponents is
H = nk=1Π((50-2k)/2) -: kor alternately H = (50/2n) x (2n-1)!!
where (2n − 1)!! (!! is the double factorial operator) is the number of ways to distribute 2n cards between n hands of two cards each. The following table shows the number of hand combinations for up to nine opponents.
Opponents
Number of possible hand combinations
1
1,225
2
690,900
3
238,360,500
4
56,372,258,250
5
≈9.7073 × 1012 (more than 9.7 trillion)
6
≈1.2620 × 1015 (more than 1.2 quadrillion)
7
≈1.2674 × 1017 (more than 126 quadrillion)
8
≈9.9804 × 1018 (almost 10 quintillion)
9
≈6.2211 × 1020 (more than 622 quintillion)
An exhaustive analysis of all of the match ups in Texas Hold ‘em of a player against nine opponents requires evaluating each possible board for each distinct starting hand against each possible combination of hands held by nine opponents, which is
69 x (50/18) x 17!! x (32/5) ~ 2.117 x 1028 (more than 21 octillion.)
If you were able to evaluate one trillion (1012) combinations every second, it would take over 670 million years to evaluate all of the hand/board combinations. While it is possible to significantly reduce the total number of combinations by pruning combinations with identical properties, the total number of situations is still well beyond the number that can be evaluated by brute force. For this reason, most software programs compute probabilities and expected values for Hold ‘em poker hands against multiple opponents by simulating the play of thousands or even millions of hands to determine statistical probabilities.
Gambling isn’t in the cards for Ohio Gov. Ted Strickland’s budget, as far as House Democrats are concerned. Their budget bill, a revision of the governor’s plan, doesn’t include a proposal to put lottery-run slot machines at horse racing tracks, said Keary McCarthy, spokesman for House Speaker Armond Budish, on Monday.
A confidence trick, confidence game, also known as a con, scam, grift or flim flam, is an attempt to intentionally mislead a person or persons (known as the “mark”) usually with the goal of financial or other gain.The confidence trickster, con man, grifter, scam artist or con artist often works with one or more accomplices called shills, who try to encourage the mark by pretending to believe the trickster. In a traditional con, the mark is encouraged to believe that they will obtain money dishonestly by cheating a third party, and is stunned to find that due to what appears to be an error in pulling off the scam they are the one who loses money; in more general use, the term con is used for any fraud in which the victim is tricked into losing money by false promises of gain.
Most confidence tricks exploit the greed and dishonesty of their victims. Often, the mark tries to out-cheat the con artist, only to discover that the mark has been manipulated into this from the start. This is such a general principle in confidence tricks that there is a saying among conmen that “you can’t cheat an honest man.”
However, some tricks depend on the honesty of the victim. A common scam is where, as part of an apparently legitimate transaction, the victim is sent a worthless check, which the victim then deposits. The victim is then urged to forward the value of the check to the trickster as cash, which they may do before discovering the check bounces. A fashionable scenario, (as of 2006), is where the victim is recruited as a “financial agent” to collect “business debts”. Paper checks are not always involved: funds may be transferred electronically from another victim. More detail can be found at ScamSpeak.
Sometimes con men rely on naïve individuals who put their confidence in get-rich-quick schemes, such as ‘too good to be true’ investments. It may take years for the wider community to discover that such ‘investment’ schemes are bogus, and usually it is too late, as many people have lost their life savings in something they have been confident of investing in.
Quotations
In response to the question “Who’s going to believe a con artist?” Ben Matlock of Matlock responded, “Everyone, if she’s good.”
References
Blundell, Nigel [1982] (1984). The World’s Greatest Crooks and Conmen and other mischievous malefactors. London: Octopus Books. ISBN 0-7064-2144-2.
Maurer, David W. (1940). The Big Con: The Story of the Confidence Man and the Confidence Game. New York: The Bobbs Merrill company. ISBN 0-3854-9538-2.
Maurer, David W. (1974). The American Confidence Man. Springfield: Charles C. Thomas, Publisher. ISBN 0-3980-2974-1.
Ball, J. Bowyer; Whaley, Barton (1982). Cheating and Deception. New Brunswick (USA), London (UK): Transaction Publishers. ISBN 0-88738-868-X.
The Alabama Supreme Court on Friday allowed the Governor’s Task Force on Illegal Gambling to keep machines and money seized in a raid last month on a White Hall gambling center and cleared the way for another raid if the task force chooses.
A scratchcard (also called scratch off, scratch game, scratch ticket, scratcher or scratchie) is a small piece of card where an area has been covered by a substance that cannot be seen through, but can be scratched off. Under this area are concealed the items/pictures that must be ‘found’ in order to win. It is a form of gambling.Simple scratchcards require the player to match three (or varying numbers) of the same prize amounts. If this is accomplished, they win that amount. More complicated scratchcards have several different ways to win on one card. Other scratchcards involve matching symbols, pictures or words, or are adaptions of popular (card-)games like Blackjack, Poker or Monopoly.
There are also online versions of the same game which utilize Flash and Java to simulate the experience on a computer.
Scratchcards are a very popular form of gambling due to their low cost. You can also win instantly as opposed to waiting for a drawing like many lotteries.
The low cost to buy a scratchcard is offset by the smaller prizes and lower expected value, compared to casino jackpots or lottery wins. There is a trend towards more expensive scratchcards (20-30USD) that have prizes in the millions of dollars.
The other recent change in scratchcards is the increasing attempts to predict which cards have prizes. In the past, scratchcards have been a random impulse purchase. Now, there are amateur and professional efforts to track the number of prizes won and cards sold so people have access to the current odds on all the games available.
Some people oppose gambling from a moral standpoint. However, in no place in the Bible does God say that gambling is wrong. In fact, as gambling is defined in Alabama, it seems to me would apply to drawing lots. That is how the apostle Matthias was chosen to take the place of Judas: “And they cast their lots and the lot fell with Matthias. And he was numbered with the eleven apostles.” (Acts 1:26)
Photo finish of a greyhound race in Tampa, Florida, USA on February 9, 1939.
Greyhound racing is the sport of racing greyhounds. The dogs chase a lure (an artificial hare or rabbit) on a track until they arrive at the finish line. The one that arrives first is the winner.
In many countries, greyhound racing is purely amateur and conducted for enjoyment. In other countries (particularly the US, UK, and Australia), greyhound racing is a popular form of parimutuel gambling, similar to horse racing. There is some popular concern in the latter countries regarding the well-being of the dogs; the effectiveness of industry efforts to address these concerns is controversial. A greyhound adoption movement has arisen to assist retired racing dogs in finding homes as pets.
Nine gaming measures have been scheduled for public hearing before Maine lawmakers. Gov. John Baldacci, a critic of expanded gambling in Maine, says his basic stance remains the same.
Play begins with each player being dealt one card face down, followed by one card face up (beginning as usual with the player to the dealer’s left). If played with a bring-in, the player with the lowest-ranking upcard must pay the bring in, and betting proceeds after that. If two players have equally ranked low cards, suit rankings may be used to break the tie. If there is no bring-in, then the first betting round begins with the player showing the highest-ranking upcard, who may check. In this case, suit should not be used to break ties; if two players have the same high upcard, the one first in clockwise rotation from the dealer acts first.
After the first betting round is complete, another face-up card is dealt to each player (after a burn card, starting with the player to the dealer’s left, as will all subsequent rounds). Betting now begins with the player whose upcards make the best poker hand (since fewer than five cards are face up, this means no straights, flushes, or full houses). On this and subsequent betting rounds, the player to act first may check or bet up to the game’s limit. The second betting round is followed by a third upcard to each player and a third betting round, again starting with the player with the best poker hand showing (thus, the first player to act on each round may change). A fourth face-up card and fourth betting round is followed by a showdown, if necessary (it usually won’t be–most deals of five-card stud end early when a player bets and gets no calls).
Here’s a sample deal. Assume that a game is being played by four players: Alice, who is dealing, Bob, who is sitting to her left, Carol to his left, and David to Carol’s left. Alice deals one card face down to each player, followed by one card face up to each player, beginning with Bob and ending with herself. Bob is dealt the ♠, Carol the K♦, David the ♦, and Alice the ♣. Because they had earlier agreed to play with a $1 bring-in, David is required to start the betting with a $1 bring-in (his ♦ is lower than Bob’s ♠ by suit). He has the option to open the betting for more, but he chooses to bet only the required $1. The bring-in sets the current bet amount to $1, so Alice cannot check. She decides to call. Bob folds, indicating this by turning his upcard face down and discarding his cards. Carol raises to $3. David folds (forfeiting his bring-in), and Alice calls. Alice now deals a second face-up card to each remaining player: Carol is dealt the J♣, and Alice the K♥. Alice’s two face-up cards make a poker hand of no pair, K-9 high, and Carol has K-J high, so it is Carol’s turn to bet. She checks, as does Alice, ending the betting round. Another face up card is dealt: Carol gets the ♥, and Alice gets the K♣. Alice now has a pair of kings showing, and Carol still has no pair, so Alice bets first. She bets $5, and Carol folds. Alice wins the pot without a showdown.
Massachusetts Senate President Therese Murray pulled an imaginary slot machine arm and proclaimed “cha-ching” on Wednesday as she revealed she, House Speaker Robert DeLeo and Gov. Deval Patrick have agreed to pass some sort of gambling-expansion bill this fall.
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Photo finish of a greyhound race in Tampa, Florida, USA on February 9, 1939.
