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Poker Overview 06: Odds and Probabilities

The relative values of Poker hands were not just conjured up by some rule maker or arbitrarily assigned by the first Poker players. They were discovered through the use of permutation and combination formulas. The exact number of possible five-card Poker hands in a 52-card deck is 2,598,960.

These hands were divided into group (ranks) of : no pair, pair, three of a kind, straight, flush, full house, four of a kind, straight flush and royal flush. The ranks were then arranged in relative value according to the frequency of their occurrence. The hands which can be expected to appear most often have the lower rank; those which appear least often the highest rank. See the ranking of hands here.

A good poker player must have a fair idea of Poker odds and probabilities. Without such knowledge, he has no good way of deciding on his course of action in the various situations which arise. That is, he has no way of making a mathematical analysis on which to base a decision. The tables that follow provide the information that will help a poker player make an analysis.

These same tables can also be used to prove the relative value of Poker hands and to settle disputes that arise regarding the chances of drawing certain valuable hands in Five-Card Draw Poker or in the first five cards dealt in any other form of Poker.

It would be simple if all one had to do to become a winning player was to memorize the following Poker tables. Knowing the exact strength of your hand or the exact chances of bettering your hand on the draw will not always help you. Thats because the playing habits of your opponents will often throw a monkey wrench into your best-laid mathematical plans. Example: A big raise from a habitually tight player means quite a different thing from the same big raise from a drunk who has already been caught trying to steal (bluff) the last half-dozen pots.

Although Poker is a game of skill, the judgments and decisions to be made by even the average Poker player involve a general knowledge of the game's probabilities.

The chances of being dealt any certain pat hand are the same, regardless of the number of players in the game. The same hold true in drawing cards to try to improve a hand. There are 2,598,960 different possible poker hands in a 52-card deck.

The following table lists the name of each different possible hand in order of their rank. It also lists the possible number of ways each can be made and the chances of being dealt such a hand in the first five cards dealt. An example would be the original five cards dealt in Five-Card Draw Poker before you draw.

Table 1
Rank of Hands Number of Possible Ways Hand can be Made Chance of Being Dealt in Original 5 Cards
Royal Flush 4 1 in 649,740.00
Straight Flush 36 1 in 72,193.33
Four of a Kind 624 1 in 4,165.00
Full House 3,744 1 in 694.16
Flush 5,108 1 in 508.80
Straight 10,200 1 in 254.80
Three of a Kind 54,912 1 in 47.32
Two Pairs 123,552 1 in 21.03
One Pair 1,098,240 1 in 2.36
No Pair Hand 1,302,504 1 in 1.99
     
TOTAL 2,598,960  

In the chance column above, fractional figures have been carried out to only two decimal places, since further extension would mean little.

The probability of being dealt a pair or better in the first five cards dealt is almost even—to be exact, 0.499--- and the probability of being dealt a no pair hand is practically the same — 0.501. So it's almost a 3 to 1 chance, when playing against two opponents, that one of them will hold a pair or better in the first five dealt cards. The probabilities vary slightly depending upon what you hold.

The 1,302,540 possible five-card no-pair hands are divided as follows:

Possible Poker Hands of Less Value Than One Pair

Table 2
Ace Counting High King Counting High, Ace Low Number of Possible No-Pair Hands
Ace hgh King high 502,860
King high Queen high 335,580
Queen high Jack high 213,180
Jack high Ten high 127,500
Ten high Nine high 70,380
Nine high Eight high 34,680
Eight high Seven high 14,280
Seven high Six high 4,080
     
  TOTAL 1,302,540

The lowest-ranking regular five-card Poker hand is comprised of 7,5,4,3,2 in mixed suits. The above table is particularly helpful to players who play high-low variants of Poker. In the short run each additional active player in the game increases the odds against you on any particular hand. In the long run, since all players have to put an equal sum into the pot, thus increasing the size of the pot in direct ratio to the increased odds, it doesn't make much difference as far as odds are concerned if you are bucking one or seven players. In High-Low Poker, where aces count both high and low, the perfect low hand is 6,4,3,2, and ace.

To simplify matters, the figures in the following two tables have been rounded out when necessary to the nearest 1/2 or whole number.

Note that your chances of making four of a kind are three times as great when drawing a pair minus a kicker than when holding a kicker.

Odds Against Improving the Hand in Draw Poker When Drawing Three Cards to One Pair

Table 3
Odds against any improvement 2.5 to 1
Odds against making two pairs 5 to 1
Odds against making three of a kind 8 to 1
Odds against making a full house 97 to 1
Odds against making four of a kind 359 to 1

Odds Against Improving the Hand in Draw Poker When Drawing Two Cards to a Pair and a Kicker

Table 4
Odds against any improvement 3 to 1
Odds against making two pairs 5 to 1
Odds against making three of a kind 12 to 1
Odds against making a full house 119 to 1
Odds against making four of a kind 1,080 to 1

In fact, you have a better chance of improving your hand when drawing three cards to a pair than when drawing two cards to a pair plus a kicker. The tables above give ample proof of that. However, good Poker playing demands that a player occasionally hold a kicker with a pair so as to keep your opponents in doubt as to your playing habits.

The odds against making a full house by drawing one card to two pairs are about 11 to 1.

Odds Against Chances of Improving the Hand in Draw Poker When Drawing Two Cards to Three of a Kind

Table 5
Odds against any improvement 8.5 to 1
Odds against making a full house 15.5 to 1
Odds against making four of a kind 22.5 to 1

Chances of Improving the Hand in Draw Poker When Drawing One Card to Three of a Kind Plus a Kicker

Table 6
Odds against any improvement 11 to 1
Odds against making a full house 15 to 1
Odds against making four of a kind 46 to 1

These two tables above show that the best chance for improvement with three of a kind is to draw two cards and not hold a kicker. Holding a kicker increases the odds against the player for any improvement.

Chances of Odds Against Filling in a Four-Card Straight in Draw Poker

Table 7
Odds against making a straight open at one end 11 to 1
Odds against making a straight open in the middle 11 to 1
Odds against making a straight open at both ends 5 to 1

Odds Against Filling in a Four-Card Flush in Draw Poker

The odds against making a flush by drawing one card of the same suit are about 4.5 to 1. If you insist on drawing to a three-card flush, the odds against your catching two cards of the same suit are approximately 23 to 1.

Odds Against Making a Straight Flush in Draw Poker

When drawing one card to a four-card straight flush, which may be open in the middle, at one end, or both ends.

Table 8
Odds against making a straight flush open at one end 46 to 1
Odds against making a straight flush open in the middle 46 to 1
Odds against making a straight flush open at both ends 22 to 1

The odds against making a royal flush are the same as a straight flush in similar conditions.

Chances of Holding Various Poker Hands in the First Five Cards Dealt When the Joker is Wild Making a 53-Card Pack

Table 9
Rank of Hands Number of Possible Ways Hand can be Made Chance of Being Dealt in Original 5 Cards
Five of a Kind 13 1 in 220,745.0
Royal Flush 24 1 in 119,570.2
Straight Flush 216 1 in 13,285.5
Four of a Kind 3,120 1 in 919.7
Full House 6,552 1 in 437.9
Flush 7,768 1 in 369.3
Straight 20,532 1 in 139.7
Three of a Kind 137,280 1 in 20.9
Two Pairs 123,552 1 in 23.2
One Pair 1,268,088 1 in 2.26
No-Pair Hand 1,302,540 1 in 2.20
     
TOTAL 2,869,685  

A very unusual mathematical situation arises in Joker Wild regarding the relative value of three of a kind and two pairs. As you see above, the chances of drawing three of a kind are one in 20.9 and the chances of drawing tow pairs are one in 23.2. Since there is a better chance of drawing three of a kind than two pairs the latter should be of a higher rank and beat three of a kind.

This peculiar situation is caused by the fact that there are 82,368 possible five-card Poker hands that contain a pair plus the joker. When a player holds one of these 82,368 hands he values his hand at three of a kind, making the joker count the same denomination as the pair he is holding. But if we should permit the two pairs to rank as a higher hand than three of a kind, we would accomplish nothing; the playing holding one of the 82,368 possible Poker hands containing a pair plus the joker would use the joker or form another pair with the highest-ranking odd card and would value his hand at tow pairs. This would bring the total number of two pairs in a pack of 53 cards to 205,920 and the possible number of three of a kinds to 54,912. Under these conditions, considering the relative value of both hands, three of a kind must remain of higher value than two pairs.

There are two solutions to this. I give them here in case there are a few players who may want to play a completely sound mathematical game. If the following two rules are used they will mathematically permit three of a kind to retain its higher value in relation to two pairs. I did not incorporate either rule in my laws of Joker Wild in Scarne on Cards because I have learned that you cannot change playing habits that easily.

  1. If a player has a hand containing a pair plus the joker, the joker cannot be considered wild, and therefore carries no value whatever; and the joker must be considered an odd card; or,
  2. If a player has paired aces, kings, queens, jacks, tens, nines, or eights plus the joker, he is permitted to rank his hand as three of a kind. If he has a pair ranking lower than eights plus the joker, he cannot value his hand at more than two pairs.

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