Q:

THE PROBABILITY OF A FLUSH A poker player holds a flush when all 5 cards in the hand belong to the same suit. We will find the probability of a flush when 5 cards are dealt. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck.

A) We will concentrate on spades. What is the probability that the first card dealt is a spade? What is the conditional probability that the second card is a spade, given that the first is a spade? Continue to count the remaining cards to find the conditional probabilities of a spade on the third, the fourth, and the fifth card, given in each case that all previous cards are spades.

I answered this one, no problem.... 0.25, 0.24, 0.22, 0.20, 0.19.

B) The probability of being dealt 5 spades is the product of the five probabilities you have found. Why? What is this probability?

I found out the probability of this - which is 0.0005016.9 - but why do we multiply to get this answer?

C) The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades. What is the probability of being dealt a flush?

Do I multiply the probability of getting dealt the 5 spades by 4 because there are 4 suits to possibly get a flush with?

0.0005016.9*4=0.0020064

?

Thanks so much in advance.... more questions to come

The probability of being dealt a royal flush can be calculated based on the number of royal flushes divided by the total number of poker hands. It can be formed four ways (one way for each suit), meaning odds of around 1 in 30,940 overall – or just 1 in 649,740 of forming a royal flush with your first five cards. Thanks for stopping in to check out this page about poker probability and the odds of being dealt a royal flush when playing online poker and casino poker. If you’re planning to play poker soon, don’t miss our complete guide to real money poker. In it, you’ll find all sorts of helpful information, including terminology, strategies, and so. Where P f is the probability of any type of flush, P sf is the probability of a straight flush, and P of is the probability of an ordinary flush. Bottom line: In stud poker, the probability of an ordinary flush is 0.0019654. On average, it occurs once every 509 deals. A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Flush (5 nonconsecutive cards each of the same suit). I am completely unfamiliar with poker, and just learning the principles of probability. In poker, a flush is when all five cards are the same suit. Find the probability of being dealt a flush (when being dealt five cards). A) What is the probability that the first card dealt is a club?

This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.

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**Preliminary Calculation**

Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.

*These are the same hand. Order is not important.*

The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.

The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.

Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is

This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.

The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.

If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.

Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.

Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:

One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.

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**The Poker Hands**

Here’s a ranking chart of the Poker hands.

The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.

Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.

The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.

*Definitions of Poker Hands*

## Find The Probability Of Being Dealt A Flush In Poker Meaning

Poker Hand | Definition | |
---|---|---|

1 | Royal Flush | A, K, Q, J, 10, all in the same suit |

2 | Straight Flush | Five consecutive cards, |

all in the same suit | ||

3 | Four of a Kind | Four cards of the same rank, |

one card of another rank | ||

4 | Full House | Three of a kind with a pair |

5 | Flush | Five cards of the same suit, |

not in consecutive order | ||

6 | Straight | Five consecutive cards, |

not of the same suit | ||

7 | Three of a Kind | Three cards of the same rank, |

2 cards of two other ranks | ||

8 | Two Pair | Two cards of the same rank, |

two cards of another rank, | ||

one card of a third rank | ||

9 | One Pair | Three cards of the same rank, |

3 cards of three other ranks | ||

10 | High Card | If no one has any of the above hands, |

the player with the highest card wins |

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**Counting Poker Hands**

*Straight Flush*

Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.

*Four of a Kind*

There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.

*Full House*

Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is

## Find The Probability Of Being Dealt A Flush In Poker Calculator

Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?

*Flush*

There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.

*Straight*

There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.

*Three of a Kind*

There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.

*Two Pair and One Pair*

These two are left as exercises.

*High Card*

The count is the complement that makes up 2,598,960.

The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.

*Probabilities of Poker Hands*

Poker Hand | Count | Probability | |
---|---|---|---|

2 | Straight Flush | 40 | 0.0000154 |

3 | Four of a Kind | 624 | 0.0002401 |

4 | Full House | 3,744 | 0.0014406 |

5 | Flush | 5,108 | 0.0019654 |

6 | Straight | 10,200 | 0.0039246 |

7 | Three of a Kind | 54,912 | 0.0211285 |

8 | Two Pair | 123,552 | 0.0475390 |

9 | One Pair | 1,098,240 | 0.4225690 |

10 | High Card | 1,302,540 | 0.5011774 |

Total | 2,598,960 | 1.0000000 |

## Find The Probability Of Being Dealt A Flush In Poker Rules

## Find The Probability Of Being Dealt A Flush In Poker Chart

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2017 – Dan Ma