# Find The Probability Of Being Dealt A Flush In Poker

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

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- Find The Probability Of Being Dealt A Flush In Poker Meaning
- Find The Probability Of Being Dealt A Flush In Poker Rules

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? 2 suited cards should improve to a completed flush on the flop. Probability of improving on the turn. Simple odds and outs. On the turn: Note: If you want to calculate the probability that the even will not take place, you must subtract the result from 1. Probability of improving on the river. Again, simple odds and outs. On the river: 10. The 2nd card must be one of the 3 cards that match the value of your other card. There are 51 cards left in the deck now, so the probability of being dealt a pocket pair is 3/51 = 1/17 = 5.88%. Another way to think about this is that you should be dealt a pocket pair, on average, once every 17 hands. The odds of being dealt a natural royal flush are 1 in 649,740 in any 52-card video poker game. If I know the variance on a game of video poker, how do I figure out the bankroll I would need to have a 90%-95% probability of avoiding ruin?

for various wild card specifications

Including a “Pai Gow” (“Bug”) Joker

The tables below show the probabilities of being dealt various poker hands with different wild card specifications. Each Poker hand consists of selecting the 5 best cards from a random 7 card deal.

While probabilities for the best 5 card hand from a deal of 7 cards (but no wild cards) can be calculated via direct combinatorics, the introduction of wild cards greatly complicates the combinatoric calculations. Thus, to produce the results shown here, the author wrote a computer program that would generate all possible poker hands. Each of these poker hands was evaluated for matched ranks (pairs, 3 of a kind, etc.), straights, and flushes. Wild cards introduce multiple evaluations for a given hand, and the best standard evaluation for any given hand is used in the tables.

Data from this page may be freely used provided it includes an acknowledgement to the author.

7 card poker probabilities if there are no wild cards

(Computer program and data by Bill Butler)

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

Poker Hand Nbr. of Hands Probability----------------------------------------------------

5 of a kind 0 0.00000000

Royal straight flush 4,324 0.00003232

Other straight flush 37,260 0.00027851

4 of a kind 224,848 0.00168067

Full House 3,473,184 0.02596102

Flush 4,047,644 0.03025494

Ace high straight 747,980 0.00559093

Other straights 5,432,040 0.04060289

3 of a kind 6,461,620 0.04829870

2 pairs 31,433,400 0.23495536

One pair >= Jacks 18,188,280 0.13595201

One pair <= Tens 40,439,520 0.30227345

Ace high 12,944,820 0.09675870

King high 6,386,940 0.04774049

Queen high 2,719,500 0.02032746

Jack high 963,480 0.00720173

Ten high 248,640 0.00185851

Nine high 31,080 0.00023231

Subtotals high card only 23,294,460 0.17411920

Total = 133,784,560 1.00000000

= COMBIN(52,7)

(Interesting observation: If a hand evaluates to just one pair, it is not distributed 4/13 “Jacks or better”. If you have a single middle-sized pair, you have a slightly increased chance of also having a straight which evaluates to a better hand. Thus a middle-sized pair occurs slightly less often than a high (Jacks or better) or a low (5’s or lower) pair.)

7 card poker probabilities if one “Pai Gow” (“Bug”) Joker is added to the deck

A “Pai Gow” (“Bug”) Joker is partially wild. If you are using it to complete a straight and/or a flush, it is an ordinary wild card. If you are using it for pairs, 3-of-a-kind, etc., it is forced to be an Ace.

(Computer program and data by Bill Butler)

Poker Hand Nbr. of Hands Probability

----------------------------------------------------

5 Aces 1,128 0.00000732

Royal straight flush 26,132 0.00016953

Other straight flush 184,832 0.00119909

4 of a kind 307,472 0.00199472

Full House 4,188,528 0.02717299

Flush 6,172,088 0.04004129

Ace high straight 1,554,156 0.01008255

Other straights 9,681,872 0.06281094

3 of a kind 7,470,676 0.04846585

2 pairs 35,553,816 0.23065464

One pair >= Jacks 19,273,104 0.12503386

One pair <= Tens 44,948,856 0.29160476

Ace high 14,430,780 0.09361938

King high 6,386,940 0.04143514

Queen high 2,719,500 0.01764270

Jack high 963,480 0.00625056

Ten high 248,640 0.00161305

Nine high 31,080 0.00020163

Subtotals high card only 24,780,420 0.16076246

Total = 154,143,080 1.00000000

= COMBIN(53,7)

7 card poker probabilities if one ordinary Joker is added to the deck

(Computer program and data by Bill Butler)

Poker Hand Nbr. of Hands Probability

----------------------------------------------------

5 of a kind 14,664 0.00009513

Royal straight flush 26,132 0.00016953

Other straight flush 184,832 0.00119909

4 of a kind 1,121,024 0.00727262

Full House 5,997,144 0.03890635

Flush 6,027,224 0.03910149

Ace high straight 1,543,460 0.01001316

Other straights 9,540,480 0.06189366

3 of a kind 13,315,300 0.08638273

2 pairs 31,433,400 0.20392352

One pair >= Jacks 21,170,640 0.13734408

One pair <= Tens 40,474,320 0.26257630

Ace high 12,944,820 0.08397925

King high 6,386,940 0.04143514

Queen high 2,719,500 0.01764270

Jack high 963,480 0.00625056

Ten high 248,640 0.00161305

Nine high 31,080 0.00020163

Subtotals high card only 23,294,460 0.15112232

Total = 154,143,080 1.00000000

= COMBIN(53,7)

7 card poker probabilities if two Jokers are added to the deck

(Computer program and data by Bill Butler)

Poker Hand Nbr. of Hands Probability

----------------------------------------------------

5 of a kind 88,608 0.00050033

Royal straight flush 91,764 0.00051815

Other straight flush 548,196 0.00309539

4 of a kind 3,134,544 0.01769923

Full House 8,521,104 0.04811449

Flush 8,397,324 0.04741557

Ace high straight 2,531,540 0.01429436

Other straights 14,181,120 0.08007383

3 of a kind 20,216,380 0.11415198

2 pairs 31,433,400 0.17748899

One pair >= Jacks 24,153,000 0.13638014

One pair <= Tens 40,509,120 0.22873513

Ace high 12,944,820 0.07309305

King high 6,386,940 0.03606392

Queen high 2,719,500 0.01535568

Jack high 963,480 0.00544030

Ten high 248,640 0.00140395

Nine high 31,080 0.00017549

Subtotals high card only 23,294,460 0.13153239

Total = 177,100,560 1.00000000

= COMBIN(54,7)

7 card poker probabilities with One-eyed Jacks wild

(Computer program and data by Bill Butler)

Poker Hand Nbr. of Hands Probability

----------------------------------------------------

5 of a kind 75,072 0.00056114

Royal straight flush 54,508 0.00040743

Other straight flush 447,946 0.00334826

4 of a kind 2,552,718 0.01908081

Full House 6,733,344 0.05032975

Flush 6,388,172 0.04774970

Ace high straight 1,404,464 0.01049795

Other straights 11,201,130 0.08372513

3 of a kind 15,758,140 0.11778743

2 pairs 23,810,436 0.17797596

One pair >= Jacks 16,255,890 0.12150797

One pair <= Tens 32,047,590 0.23954625

Ace high 9,743,580 0.07283038

King high 4,662,000 0.03484707

Queen high 1,888,110 0.01411306

Jack high 481,740 0.00360086

Ten high 248,640 0.00185851

Nine high 31,080 0.00023231

Subtotals high card only 17,055,150 0.12748220

Total = 133,784,560 1.00000000

= COMBIN(52,7)

7 card poker probabilities with Deuces (2’s) wild

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

(Computer program and data by Bill Butler)Poker Hand Nbr. of Hands Probability

----------------------------------------------------

5 of a kind 609,760 0.00455778

Royal straight flush 399,484 0.00298602

Other straight flush 1,552,732 0.01160621

4 of a kind 7,504,920 0.05609706

Full House 9,421,824 0.07042535

Flush 7,993,600 0.05974979

Ace high straight 4,033,160 0.03014668

Other straights 15,355,640 0.11477887

3 of a kind 20,151,920 0.15062964

2 pairs 19,491,840 0.14569574

One pair >= Jacks 16,211,160 0.12117362

One pair <= Tens 20,708,880 0.15479275

Ace high 6,386,940 0.04774049

King high 2,719,500 0.02032746

Queen high 963,480 0.00720173

Jack high 248,640 0.00185851

Ten high 31,080 0.00023231

Nine high 0 0.00000000

Subtotals high card only 10,349,640 0.07736050

Total = 133,784,560 1.00000000

= COMBIN(52,7)

7 card poker probabilities with 2 Jokers,

One-eyed Jacks, and Deuces (2’s) wild

(8 out of 54 cards are wild)

(Computer program and data by Bill Butler)

Poker Hand Nbr. of Hands Probability

----------------------------------------------------

5 of a kind 5,496,072 0.03103362

Royal straight flush 1,821,704 0.01028627

Other straight flush 6,959,976 0.03929957

4 of a kind 23,628,576 0.13341898

Full House 12,751,424 0.07200104

Flush 13,497,668 0.07621471

Ace high straight 6,037,238 0.03408932

Other straights 25,527,008 0.14413849

3 of a kind 28,206,968 0.15927091

2 pairs 14,381,496 0.08120525

One pair >= Jacks 15,378,900 0.08683711

One pair <= Tens 16,024,260 0.09048114

Ace high 4,693,080 0.02649952

King high 1,911,420 0.01079285

Queen high 629,370 0.00355374

Jack high 124,320 0.00070197

Ten high 31,080 0.00017549

Nine high 0 0.00000000

Subtotals high card only 7,389,270 0.04172358

Total = 177,100,560 1.00000000

= COMBIN(54,7)

Alsoplease see 5 card Poker probabilities

Alsoplease see 6 card Poker probabilities

Alsoplease see 8 card, 9 card, and 10 card Poker probabilities

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

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