What is The Probability of Drawing Straight In Poker?

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Math is not the exclusive domain of mathematicians and scientists, but impacts our everyday life. One such interesting example is that of math in poker. Did you know that integrating math's into your poker game will significantly enhance the ‘probability’ of you winning? Has your interest been piqued? This helps you understand things like finding fold equity, probability of a straight in poker and the odds of flopping a straight or other such hands. Read on to know more!

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How to compute poker probabilities?

Poker involves a host of complex mathematical calculations based on probabilities. A good knowledge of numbers provides an essential basis for being a good poker player. A good place to begin would be to question the probability of getting a particular hand, like odds of flopping a straight. Though there are several online sites to do this for you, it would be interesting to learn how it is done actually.

A brief overview of the same is as follows –

·         One must count the number of possible five card hands that can be dealt from a standard deck of 52 cards, following which one must count the number of ways that a particular type of poker hand can occur.

·         The probability of being dealt any particular type of hand is equal to the number of ways it can occur divided by the total number of possible five card hands.

Or,

Number of ways a particular hand can occur/ Possible five-card hands

·         The possible five card hands that one can be dealt with is calculated by using the combinations formula (i.e. the number of combinations of n objects taken r at a time)

nCr = n (n - 1) (n - 2) . . . (n - r + 1)/ r! = n! / r! (n - r)!

·         For example, we have a standard deck of 52 cards (therefore, n = 52) and we wish to arrange them in unordered groups of 5, therefore r = 5. The number of combinations thus stands at 2598960, after applying the formula. Say we wish to count the number of ways in which a royal flush was to occur i.e. a sequence of 10, Jack, Queen, King and Ace of the same suit. The number of ways that one can get a royal flush is 4 (as there are only 4 suits). Therefore, the probability of getting a royal flush is:4/ 2598960 =0.000 001 539

Rankings and probability of poker hands

In order to make one’s life easy, a table has been prepared for reference with the rankings, frequency and probability of poker hands. Peruse below –

Type of Hand

Description

No. of Ways in which these hands can occur

Probability

Royal Flush

Consists of 10, J, Q, K, A of the same suit.

4

0.000002

Straight Flush

A straight flush is five cards in sequential order belonging to the same suit. An example of a straight flush is 3, 4, 5, 6, 7, all diamonds.

36

0.000015

Four of a Kind

Consists of four cards of the same denomination. For example, 4 kings and any other card.

624

0.000240

Full House

Consists of three cards of one denomination and two cards of any another denomination. For example, holding three aces and two kings is a full house.

3744

0.001441

Flush

All the five cards are from the same suit (excluding the royal and straight flushes). For example, 2, 3, 6, 9, J (all clubs) is a flush.

5108

0.001965

Straight

The 5 cards are in sequential order, but need not belong to the same suit. For example, 3, 4, 5, 6, 7 (of any suit) is a Straight.

10200

0.003925

Three of a Kind

Consists of three cards of the same denomination. For example, a hand with three aces, one jack and one queen.

54912

0.021129

Two Pairs

Consists of two sets/ pairs of cards. For example, 3, 3, Q, Q, 5.

123552

0.047539

One Pair

Consists of one set/ pair of cards. For example, 10, 10, 4, 6, K.

1098240

0.422569

Nothing

For example, 3, 6, 8, 9, K (at least two different suits).

1302540

0.501177

What is a straight in poker?

A straight is a hand that contains five cards of sequential rank. These cards do not have to belong to the same suit. For example, seven-high straight - a hand that has spade 7, club 6, heart 5, heart 4 and a diamond three or a combination of the like. A straight hand ranks sixth in the list and places just above a three of a kind.

The highest possible straight hand is a Broadway which consists of an ace, king, queen, jack and ten. The lowest straight hand is a Wheel or a Bicycle i.e. a hand which has ace, deuce, three, four and five.

What is the probability of straight in poker?

In a standard card deck of 52 cards, there are 10200 straight hand combinations that are possible and 10 distinct ranks of straights that can be seen. The probability of a straight in poker is 0.003925.

The odds of flopping a straight in Texas Hold’em is as follows -

Pre-flop

0.39%

(based on 5 cards randomly drawn from a full 52-card deck) (excl. royal and straight flushes)

Flop

1.30%

(when holding 2 connected cards J-10 through 5-4)

Turn

16.90%

(with an open-ended-straight-draw on flop)

River

17.20%

(from an open-ended-straight-draw)

Please note: These numbers will vary with each variant of the game and must be accounted for while approaching the specific variant of poker.

What does this mean for a poker player?

Straights are usually quite strong in a Texas Hold’em game, but their strength can vary greatly based on the board texture. Straights are always the strongest hand to hold when there is no pair on the board and the chances of having a flush or higher straight is slim.

For a player, an ordinary straight is a rare event. It only happens to occur once in every 255 deals. However, these numbers are not the sole determinants of success in the game of poker – experience and intuition also play a huge role.

Conclusion

Luck is not the only factor that determines your win and loss streaks in poker. It requires a mastery over one’s emotions, tactfully played combinations, pre-calculated, mathematically sound strategies and a risk-taking attitude. Preparation and research will make a huge difference to your game. Don’t take my word for it, go ahead and make that effort to witness the results first hand!

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