Statistics 25100
Poker hands
The various types of hands in poker,
arranged from highest to lowest, are as follows:
- a straight flush -- five cards of the same suit in sequence;
- four of a kind -- four cards of the same rank, and one
other card;
- a full house -- three cards of any one rank, and two cards
of another rank;
- a flush -- five cards of the same suit, but not in sequence;
- a straight -- five cards in sequence, but not all of the
same suit;
- three of a kind -- three cards of the same rank, and two
unmatched cards;
- two pairs -- two cards of one rank and two cards of another
rank, and one unmatched card;
- one pair -- two cards of the same rank, and three unmatched cards;
- no pair -- none of the above: five unmatched cards, not all
of the same suit and not in sequence.
An ace can be played in a straight or straight flush either
high (A, K, Q, J, 10) or low (5, 4, 3, 2, A).
The following table gives the Number of (unordered) hands of
each type, and the Chance to be dealt a hand of
that type when all hands are equally likely.
We worked out some of these entries in class. See if you can
derive the others.
Hand | Number | Chance |
Straight flush | 40 | 0.0000154 |
Four of a kind | 624 | 0.0002401 |
Full house | 3,744 | 0.0014406 |
Flush | 5,108 | 0.0019654 |
Straight | 10,200 | 0.0039246 |
Three of a kind | 54,912 | 0.0211285 |
Two pairs | 123,552 | 0.0475390 |
One pair | 1,098,240 | 0.4225690 |
No pair | 1,302,540 | 0.5011774 |
Total | 2,598,960 | 1.0000000 |
Michael J. Wichura
Last modified: Tuesday, March 27, 2007