In poker, the probability of each
type of 5 card hand can be computed by calculating the
proportion of hands of that type among all possible
hands. To gather these numbers, a basic understanding of
probability as well as that of arrangements is
necessary. In essence, in five card poker hands, we are
interested in the total number of different hands that
could be dealt at random. Therefore, we are looking for
the number of possible five card combinations from a
deck of 52 cards.
In probability, this problem is
referred to as selecting the number of different five
object combinations from a group of 52 cards.
Mathematically, we determine that by dividing 52
factorial by the product of 5 factorial and 47
factorial. Those numbers come from a specific
mathematical formula that determines the number of
different combinations that can be made. When finally
computed, we determine there are more than 2.5 million
different combinations possible (2,598,960 to be exact).
To determine the probability of a
specific hand, we must determine the number of
successful hands that can be created then divide that by
the total number of hands possible. Therefore, we come
up with the following specific probabilities for each
hand.
For the case of a straight flush, we
would find that there are only 40 such possible hands
that can made. Therefore our probability of getting a
straight flush is 40 divided by 2,598,960, or about
.0000154. If we turn that fraction over, we have our
odds which in this case are 64,973 to 1, making a
straight flush the most difficult hand to obtain and the
unchallenged winner. Moving to four of a kind, we find
there are 624 possible successful outcomes, again
dividing by our more than 2 million total hands, we get
a probability of .000240 and odds at 4,164 to 1.
Similarly we could calculate the same ratios for full
house (3,744 successful hands, a probability of .00144,
and odds of 693 to 1), flush (5,108 successful hands, a
probability of .00197, and odds of 508 to 1), straight
(10,200 successful hands, a probability of .00392, and
odds of 254 to 1), three of a kind (54,912 successful
hands, a probability of .0211, and odds of 46.3 to1),
two pair (123,552 successful hands, a probability of
.0475, and odds of 20.0 to 1), one pair (1,098,240
successful hands, a probability of .423, and odds of
1.366 to 1), and no pair (1,302,540 successful hands, a
probability of .501, and odds of 0.995 to 1).
We noted in some of our other
tutorials games that featured additional cards or decks
with cards stripped away. Again, the probabilities
suggested above are the numbers for a 52 card deck with
the odds based on a five card hand. Obviously, if 7
cards are dealt, the probabilities of each event become
larger while if fewer than 52 cards are available, the
actual ranking of hands may change. In a stripped down
deck, a flush may top a full house as with fewer cards
available in a suit it is more difficult to get five
cards of the same family.
