MATHEMATICS OF POKER

Poker Mathematics Overview

The relative values of Poker hands were determined by a rather complicated mathematical process of counting the actual number of Poker hands in the pack.

Possible Poker Hands In A 52-Card Deck

 

 

Royal Flush

4

Straight Flush

36

Four of a Kind

624

Full House

3,744

Flush

5,108

Straight

10,200

Three of a Kind

54,912

Two Pairs

123,552

One Pair

1,098,240

No Pair, less than above

1,302,540

Total

2,598,960

 

 

Possible Hands of Less Value Then One Pair

 

 

Ace Counts High

Ace Counts Low

Number of Hands

Ace High

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

70,680

Nine High

Eight High

34,680

Eight High

Seven High

14,280

Seven High

Six High

4,080

Total

 

1,302,540

 

 

Chances of Being Dealt Poker Combinations In The Original Five Cards

 

 

Royal Flush

1 in 649,740

Straight Flush

1 in 64,974

Four of a Kind

1 in 4,165

Full House

1 in 649

Flush

1 in 506

Straight

1 in 255

Three of a Kind

1 in 47

Two Pair

1 in 21

One Pair

1 in 2.5

No Pair

1 in 2

 

 

Chances of Improving A Hand On The Draw

 

 

Cards Held In Hand

Cards Drawn

Possible Improvement

Odds Against Making

One Pair

3

Two Pair

Threes

Full House

Fours

Anything

5 to 1

8 to 1

97 to 1

359 to 1

2.5 to 1

One Pair And An Ace Kicker

2

Aces Up

Another Pair

Threes

Full House

Fours

Anything

7.5 to 1

17 to 1

12 to 1

119 to 1

1080 to 1

3 to 1

Two Pairs

1

Full House

Anything

11 to 1

11 to 1

Threes

2

Full House

Fours

Anything

15.5 to 1

22.5 to 1

8.5 to 1

Threes And One Odd Card

1

Full House

Fours

Anything

14.67 to 1

46 to 1

11 to 1

Four Straight (Open-End)

1

Straight

5 to 1

Four Straight (One End Or Interior)

1

Straight

11 to 1

Four Flush

1

Flush

4.25 to 1

Four Straight Flush (Both Ends Open)

1

Straight Flush

Anything

22.5 to 1

2 to 1

Four Straight Flush (Interior Or End)

1

Straight Flush

Anything

46 to 1

3 to 1

One Ace

4

Pair of Aces

Aces Up

3 to 1

14 to 1



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