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Hands of Poker or How to evaluate card combinations

There is a lot of kinds of poker exist all over the world. The most popular are Draw Poker, Poker with Joker, Oasis Poker and three-card poker.

Draw Poker – is the standard form of Poker. If you are absolutely not aware of the rules it’s the best chance to start familiarizing. The main target of draw poker is to collect the strongest hand of 5 cards. After first lead you can refuse your cards or get the card replacement. All cards should be hidden from other players before final Showdown or when it’s absolutely necessary.

There are two things which you should know to play Draw Poker. First are hands of five-card poker and second is ranking that means order or course of the game including how and when you should stake. In this article I will try to describe all Poker Hands you should know to play.

Poker Hands

Hand of poker is collected with the best 5 cards combination and evaluated according the following list. The first is the strongest.

  1. Royal Flush: A-K-Q-J-10 all cards are the one suit
  2. Straight Flush: any five successive cards of one suit
  3. Four-of-a-kind: 4 cards of one type (e.g. four kings)
  4. Full House: 3 cards of one type and one pair
  5. Flush: any five cards of one suit
  6. Straight: any five successive cards
  7. Three-of-a-kind: 3 cards of one type (e.g. three queens)
  8. Two pairs
  9. Pair: 2 cards of one type (e.g. two jacks)
  10. Absence of pairs: five different cards of different suits

There are no wild cards in Draw poker all suits are evaluated equally.

How to play Poker? Course of Play

Each poker round starts from dealer appointment. Each player from the table becomes dealer and then gives this chance to player sitting on the right.
For example if you now distribute the cards next time the player sitting on the right will do it. Dealer gets the pack and reshuffles it. Each player brings in ante. It’s special fee which gives you opportunity to play the next game. If you don’t bring in ante you are just sitting out. These sums of money are placed in the center of the table (in the pot). When the ante has been put dealer starts to distribute the cards one by one for each player while each has five cards. Players receive their cards and evaluate their hands. Player sitting on the left of dealer starts the game. He can make a bid, pass (do not make a bid) or refuse the game and fold (take the cards off).

The next player sitting on the left does the same. If he decides to make a bid he should call this bid to confirm it. Also he can raise the bit in case he considers that the situation acquires changes. He can do it only before the dealer confirms the bet. So each player including dealer makes the personal decision. If the dealer receive amounts of bids players should call their bids to continue the game. All raises are not available now. Now players can discard all or some cards in hope they will collect the better hand.

If player requires the cards replacement the dealer should give him new cards. But the cards of course should be hidden. The next round players anew continue or stop playing (call of fold the game) and all procedures repeat again.

In the final of the game players make a Showdown. After the final round players open their hands to show cards combination they have collected. Players compare these combinations and the “best hand” receives the pot. If two players (sometimes it happens) have the equal card combinations the pot is splitting for two players 50/50.

Poker as the most “profitable” gamble

It’s not a secret for us that poker is one of the most popular games nowadays. It appeared five centuries ago nearby in XI century as scientists suppose now. Firstly the gamble appeared in Europe to be more precise in was popular in France, Italy and Spain. At that time the rules had some differences and commonly it was announcement of players’ stakes and comparison of cards combinations. Person who had the biggest combination was a winner. With the lapse of time the rules acquired changes while the modern poker appeared as large as life. It has got all recent varieties and rules.
Poker was brought in America from Europe during colonization. Here it became national gamble and here exactly people start to play it with 52 cards. Today it is one of the most popular games not only in USA but all over the world. Poker was the first gamble appeared in casinos which were then developing rapidly. Starts from the last century “club poker” was born and acquired its popularity. In that game gamblers played not versus croupier but versus each other. Today there are more than ten kinds of club poker exist which are different from each other with the insignificant features. But all it have the one aim to get the biggest combination of 3 or 5 cards.
Interesting facts:

In 1963 in the popular Las Vegas Grand Royal casino located in USA was fixed the biggest winning in Poker with Joker. Someone with name Jason Kamuela staked maximum on ante (it is $5000) and then exchange one spade Royal Flush card. He exchanges his spade ace for Joker (It should be noted that in Royal Flush Joker is exchanged only for spade ace) and wins (according payout rate 1:100) $1 000 000. People say not without purpose that of course it’s nice to win something in roulette or black jack but only in poker you can win your “assets”.

“Progressive” view on Caribbean Poker

When you’ll visit a casino next time look at Caribbean Poker tables and you’ll see that every player almost makes an additional stake on jackpot.
Majority of game analysts will say you: “By no means!”. Stake on jackpot is one of the worst, casino advantage here reachs 60, 70, even 80% subject to jackpot size. Even when jackpot becomes rather big and player’s advantage reachs, payings are so rare that your chances are imminent zero.
But when a player asks me if to make an additional stake I always answer with counter questions: “And what is your aim? What are you playing for?”.
For some players the aim of visit a casino is a decrease of casino advantage to minimum for own’s gain maximization. These players not only don’t stake on jackpot but don’t play Caribbean Poker at all. 5,2% casino advantage by optimal strategy is too much. Serious players go to play blackjack (casino advantage by basic strategy is 0,5%) or dices (casino advantage is 0,6% by combination Pass/2хOdds).
Caribbean Poker players gather around tables because of two reasons. First of all it is a light relaxing game, one of the slowest in casino. Secondly staking on jackpot gives them a chance of a big gain which can’t be get in other card games. Blackjack gives up a higher percent to a player but blackjack player never leaves a casino with $10.000, $100.000 or even $712.070 (record jackpot in Caribbean Poker) after the one one-dollar stake.
Jackpot hunters make the additional stake at the same time with the main stake. Most of Caribbean Poker tables have coin slots in front of the main game box. Player put one-dollar fish to the slot before deal. When all stakes are made a coin falls into coin slot simultaneously with deal beginning and the indicator is alight.
After all cards are dealed a player decides if to play or not. After a dealer turns up player’s cards who is in game, players who staked on jackpot and have flush or higher get a fixed sum regardless of dealer’s cards.
Additional stake winnings are pay in compliance with a table reminiscent of videoPoker. In the majority of casino $1 stake gets you $50 for flush, $75 for full-house, $100 for 4 of a kind, 10% of jackpot or $5.000 for street flush and full size of jackpot for royal flush. It is minimal payings established by Mikong Gaming company, casino can raise stated values.
Each of winning combinations is not easy to get in Caribbean Poker. So, flush occurs one time for 509 games on average, full-house one time for 694 games, 4 of a kind one time for 4165 games and royal flush one time for 649.740 games! In aggregate a winning on additional stake occurs one time for 273 games on average. You can osreve your dollars dissappearing in table’s depth during 7-8 hours before you’ll get something instead.
The more jackpot the less casino advantage. When jackpot reaches $261.950 a stake cease to be losing and a player gets an advantage before casino. But it doesn’t mean that you immediately get a perfect chance to win greately. You’ll win one time in 273 games as before. It means only that if you have staked on jackpot endlessly you didn’t come off a looser finally.
But even when staking on jackpot gives a player an advantage he still looses 5% on basic Poker stake. To make a game profitable in tote with $5 ant a jackpot value has to reach $351.200 already.
Let’s back to initial question: have a Caribbean Poker player to make an additional stake? It depends on you. When jackpot size is low casino advantage is great and winnings are rare whether or no. If you care about your money don’t stake on jackpot better. But if you prefer slots and jackpot fans point of view and ready to suffer additional losses for the sake of big gain, the Caribbean Poker its’ additional stake is right for you!

Applied probability theory

I don`t stop repeating that if you are to some extent deeply planning to play, if you hope to begin win in casino you have to know the base of probability theory at least. So I`ll try to “apply” this probability theory to gambling questions. Prob theory and games are closely related indeed. For example the French word “hazard” – “case”, “risk” came probably from Arab “azar” – “dice”. Actually the science arose from gambling questions exactly and dice game exactly!

It is considered that order holder d`Mere wrote a letter to Bles Paskal asking him how many times have to drop dices to have a probability of two sixes occurring more than ½ (ho thought that 24 drops is enough). Paskal answered this question, began to raise analogous questions in his correspondence with Ferma and these two scientists` names are usually calls as founders of probability theory (it`s about the middle of 17 century). Third “father-founder” is Guigens who also answered the players` question if during three dices dropping 11 or 12 points will occur oftener. Of course certain questions of probability theory were also raise earlier but the base of harmonious theory was laid by these famous scientists.

So, let`s give the definition of probability theory, that is the mathematics section learning random magnitudes` regularity: random events, random magnitudes, their properties and operations with them. Lets` continue… Random event is event which can happen or not in present conditions and which have a defined probability p (from 0 to 1) of its occurring. Random magnitude is magnitude which assumes one of many values as a result of experience and appearance of one or other value of this magnitude is random event at that.

Since history of science development began with dices we`ll appeal to this game, it is too opportune for these. We have dice (cube) with six sides and numbers from 1 to 6 on each. Random event will be for example each number occurring for example 1. If there is normal not “charged” cube each of sides will occur with the same frequency that is one time of six. That 1/6 is probability of our random event occurring (occurring of 1). Probability designates with letter p or P usually. And the random magnitude with our experience (dice dropping) will be occurring of the one or other number exactly. Set of probable magnitudes are 1, 2, 3, 4, 5 and 6 with it. As it can`t be other numbers occurring (there are no 7 or 8 on a cube), these numbers from 1 to six will be total event system. Event probabilities sum in the total system is 1 (that`s why if events are equal probable it is easy to calculate each event probability as 1/event quantity). As it is not possible to occur 1 and 6 at the same time when dropping cube (and other two numbers, we don`t consider “standing on a edge” and “hover in the air” cases) these events will be incompatible between each other.

I offer you the first law of prob theory, addition rule: probability of occurring in experience any one (it doesn`t matter which exactly) of results is sum of these results probabilities if each two of them are incompatibly between each other (we can write it like Р(А or В)=Р(А)+Р(В), where Р(А) is probability of event A). For example if you are interested in probability of even number occurring (that means 2, 4 or 6) we have to sum up probabilities of each of given numbers occurring, we`ll get 1/6+1/6+1/6=1/2 or 50%. Odd number occurring (1, 3 or 5) will be opposite even number occurring event. Sum of opposite events probabilities is 1, that is probability of odd number occurring is 1-1/2=1/2. Event probability is often opportune to calculate with probability of opposite event occurring. So a probability of less than 6 number occurring can be define as probability of 6 UN occurring and calculate like 1-1/6=5/6. Of course in the case of dices it is not hard to sum up 1/6 for five times but in the most of real sums it will be vastly harder.

Let`s pass on conditional probability notion. Suppose that we have two cubes: one standard with numbers from 1 to 6 and the second where 1 and 2 numbers repeated twice and 3. If we`ll drop the first cube, 1 will occur with 1/6 probability, and if we`ll drop the second we`ll have the 1/3 probability already. And 6 will occur with 1/6 and 0 probability (event with 0 probability calls impossible, and 1 – reliable) respectively. Probability of dropping some number on the first dice, for example 1, calls unconditional as it doesn`t depend on other events and probability to drop 1 on both cubes at the same time will be conditional already (some conditions have to happen at the same time – 1 on the first and the second cube). How can we detect this probability? Probabilities multiplication rule will help us: Product of two events probability is product of probability of one of them and conditional probability of other one, calculated with the condition that first event have happened already (Р(А and В)=Р(А)*РА(В), where РА(В) is B event probability when A performs). That is in our example with cubes probability of event A – 1 occurring on the first cube – is 1/6, probability of 1 occurring on the second cube, with the condition of 1 occurring on the first cube too, is 1/3. In all 1 occurring on both cubes is 1/6*1/3=1/18.

Indeed probability of 1 occurring on a second cube is 1/3 independently of what will occur on a first cube. Thereby occurring of 1 on each cube are independent events. In that case multiplication rule simplifies to Р(А and В)=Р(А)*Р(В) – probability of combined occurring any number of independent of each other events is product of probabilities of these events. Probability of two 6 or 1simultaneous occurring on both ordinary cubes is 1/6*1/6=1/36. And what is the probability of 4 or 3 occurring? Is it 1/36 too? No! For 6-6 occurring 6 have to occur on a first cube and 6 on a second. And for 3-4 it can be 3 on the first and 4 on the second or 4 on the first and 3 on the second. It happens that we have to find two conditional probabilities and then sum up them, we`ll get 1/18 probability. Armed with this knowledge you can become like Guigens and answer the question he was asked for. Find all combination which give 11 and 12 points (for example 11=1+5+5, 11=2+4+5 and so on), take into account possibility (or impossibility) of combinations occurring in different order (that is 1,5,5 or 5,1,5 or 5,5,1 – three possible combination in all). Then you multiply and sum up everything. You have to get 25 combinations (each has 1/6*1/6*1/6=1/216 probability) for 12 and 27 for 11. I`ll repeat once again – cubes droppings or numbers occurring in roulette are independent events, probabilities of each are the same and occurring of cards from one batch are dependent already. For example probability of pulling ace first from a 52-cards batch is 4/52. And probability of second ace is 3/51 already (as there are only three aces and 51 cards remained in batch). As a result probability of pulling two aces one by one is 4/52*3/51. And if the first card was not ace the probability of pulling ace with a second card was 4/51 and total situation probability was 48/52*4/51.

Indeed this given higher theory is enough for calculation of most situations in casino games. In the same roulette or dices it is more than enough for calculating of different probabilities. Let`s take “the favorite” roulette-men`s example – 10 reds one by one and calculate it`s probability. There are 37 numbers in total, 18 red among them, so the probability of red occurring is 18/37. Probability of two reds one by one is 18/37*18/37 already, and 10 one by one is (18/37)^10=0,074% or 1 / 1347 (^ – involution). If calculate not only colour but colour plus zero which is dangerous for martingale-lovers we`ll get (19/37)^10=0,13% already or 1 / 784. That is during 1000 spins at the average (one day of playing) you expect occurring of series from 10 one by one not of “your colour”. And if there are 10 of any colour one by one (we`ve counted only the one colour) a probability doubles as series can be both red and black. Let`s look for probability of not occurring of dozen 15 times running. First dozen not occurring means that only second and third dozens occur, plus zero. Probability of their occurring is 1-12/37. Probability of their 15 times running occurring is (25/37)^15=0,28% or 1 for 358 spins. But again if we are interested in not occurring of any dozens but not exactly the first one, we multiply our total by three and approach to 1 case for 100 spins. One who wishes can calculate other probabilities too as number occurring for three times running or not occurring of number during 100 spins.

So, let`s back to order holder d`Mere`s sum. We know that probability of two 6 occurring is 1/36. Probability of two 6 not occurring (opposite event) is 1-1/36=35/56. Probability of two 6 never occurring during 24 droppings is (35/36)^24=50,8%. So, two 6 occurring probability once at least is opposite event with the probability 1-50,8%=49,2%. As we can see d`Mare`s supposition was wrong, you can be convinced after Paskal that it is 25 droppings necessary for 6-6 occurring probability exceed 50%. Turn your attention that sums of “at least once happened event” optimally solve with probability of opposite events.

One more very useful function is Bernoulli formula which makes it possible to estimate probability that during N tests with probable success p amount of successes will be P=p^n*(1-p)^(N-n)*N!/n!/(N-n)! If you use Excel there is function BINOMRASP which enables to calculate respective probabilities quickly. Generally given formula enables to calculate binomial distribution coefficient. This formula is most urgent for “chances” check of one or other events. For example 5 zeros (n successes) occurred during 100 spins (N tests), occurring probability is 1/37 (success probability p). We substitute to the formula or better use BINOMRASP in Excel (“Integral” parameter set as “lie”), we`ll get 8% probability of 5 zeros occurring during 100 spins, that is quite a lot. Pay your attention that is the probability that zero will occur 5 times running! But it can occur 6 times with 3% probability and 7 times with 1,3% probability etc.  If we want to know probability of 5 and more times occurring we have to sum up these probabilities or it is handier to calculate in Excel by 1-BINOMRASP (4;100;1/37;ИСТИНА)=13,5% formula (the event opposite to zero occurring 4 or less times).

When solving different sums of applied probability theory you can need main combinatorial analysis formulas:

  • Permutations – with how many ways you can arrange in order available n elements: P=n! (I hope, you`ve got acquainted with factorial notion – that is product of all natural numbers from 1 to n. n!=1*2*3*…*(n-1)*n). When we talked about two or three dropping of cubes, we were interested exactly in permutations for two or three different numbers. Actually we multiplied probability of defined combination occurring by permutations quantity, for three cubes we have Р=3!=6 variants of permutations. If we have  some (m) same elements the formula changes to P=n!/m! (when two cubes of three are the same as 1-5-5, there will be 3!/2!=3 permutations, and if all three cubes are the same, we`ll have 3!/3!=1 permutations).
  • Placements – with how many ways you can rearrange m elements from available array to n: A=n!/(n-m)! For example how many different numbers can we compose using only three numbers of ten? 10!/(10-3)!=8*9*10=720 (the digit is less than thousand because digits like 111 or 505 use one figure more than once, for placement with reiteration the formula will be n^m, that will give us 1000 of three-digit figures). Formula of placement often uses for possible poker combinations detection. For example what is a probability to collect royal flesh with five given cards? Only 5 cards of 52 in batch can be deal with 52!/47!=311.875.200 ways (we`ll use placement formula right). Royal flesh of defined suit can be collect with 5!=120 ways (the amount of 5 cards placements which form royal). As there are only 4 suits, the probability to collect royal flesh will be 4*120/311.875.200=1/649740 or about every 650-thousand dealing.
  • Combinations – with how many ways m elements can be choose from n elements: C=n!/m!/(n-m)! For example with how many ways two preflop-cards can be deal in Texas holdem? 52!/2!/50!=51*52/2=1326 ways, when 9♣8♠ and 8♠9♣ consider as the one way, as order is not important for us. Further each pair can be compose with 4!/2!/2!=6 ways already (the quantity of combinations is in twos from four), each unpaired combination with 4^2=16 ways (4 ways to choose first card and 4 for the second, multiply analogous to probabilities), one-suit combination of them with 4 ways (by suit quantity) and each different-suit with 16-4=12 ways.

I hope, you`ll understand what is written here. For practice you can train, calculate probability of different combinations occurring in dices and sic bo and compare them with payments; calculate probability of getting different “series” in roulette; calculate probability of getting different bonus payments in different blackjack versions, and compare results for different batches quantity – 1, 2, 3, 4, 8 etc. at that. I wish you success in mathematics mastering and game!