5 card poker hands with misc. wild cards
6 card poker hands with misc. wild cards
7 card poker hands with misc. wild cards
8 card, 9 card, and 10 card poker hands with misc. wild
Lowball (Low Ball) poker probabilities with misc. wild cards
(5 to 10 cards)
Click here for optimal strategy and expected value for Video
The probability of being dealt various poker
hands has been printed in many other sources. We present the
probabilities for a 5 card deal here, and then concentrate on
how to calculate these numbers.
Hand Number of
Four of a
Three of a
5 card poker hands with misc. wild cards
The first calculation that must be made is to
determine the total possible poker hands. A poker hand
consists of 5 cards randomly drawn from a deck of 52 cards.
Thus, the number of combinations is COMBIN(52, 5) = 2,598,960.
Each of these 2,598,960 hands is equally likely. For each of
the above “Number of Combinations”, we divide by this number
to get the probability of being dealt any particular hand.
For the calculations, we will first split out the
“No Pair” hands which include Royal Straight Flushes, Straight
Flushes, Flushes, Straights, and “Nothings”.
Then, we will look at all combinations that have at
least 1 pair.
The cards in a hand without any pairs will have 5
different denominations selected randomly from the 13
available (2, 3, 4...Ace). Also, each of the 5 denominations
will select 1 suit from the four available suits. Thus the
total number of no-pair hands will equal:
COMBIN(13, 5) * (COMBIN(4, 1))^5 = 1287 * 1024 = 1,317,888.
A Straight Flush consists of 5 consecutive cards
in the same suit and may have a high card of 5, 6, 7, 8, 9,
10, Jack, Queen, King, or Ace for a total of 10 different
ranks. Each of these may be in any of 4 suits. Thus there are
40 possible Straight Flushes. An Ace high Straight Flush is a
Royal Flush. Since there are only 4 different suits, there are
only 4 possible Royal Straight Flushes. When we subtract the 4
Royal Straight Flushes from the total of 40 Straight Flushes,
we are left with 36 other Straight Flushes that are King high
A Flush consists of any 5 of the 13 cards from a
particular suit. There are 4 possible suits. Thus the number
of possible Flushes is: COMBIN(13, 5) * 4 = 5,148. However,
this includes the 40 possible Straight Flushes. When we
subtract these out, we are left with: 5,148 - 40 = 5,108
possible ordinary Flushes.
A Straight consists of 5 cards with consecutive
denominations and may have a high card of 5, 6, 7, 8, 9, 10,
Jack, Queen, King, or Ace for a total of 10 different ranks.
Each of these 5 cards may be in any of the 4 suits. Thus there
are 10 * 4^5 = 10,240 different possible straights . However,
this total includes the 40 possible Straight Flushes. Thus we
subtract 40, which leaves us with 10,200 possible ordinary
Finally, we come to the “Nothing” hands which are
basically all the left over garbage. This is simply the total
number of “No Pair” hands minus all the good stuff. This gives
us: 1,317,888 - 4 - 36 -5,108 - 10,200 = 1,302,540 “Nothing”
Now on to 1 pair or better. A hand with just 1
pair has 4 different denominations selected randomly from the
13 available denominations. 3 of these denominations will
select 1 card randomly from the 4 available suits. The 4th
denomination will select 2 cards from the available 4 suits.
Finally, the pair can be any one of the four available
denominations. Thus the calculation is: COMBIN(13, 4) *
(COMBIN(4, 1))^3 * COMBIN( 4, 2) * 4 = 1,098,240 possible
hands that have just one pair.
The calculation for a hand with two pairs is
similar. We will have 3 random denominations taken from the 13
available. Two of these denominations will use 2 of the four
available suits while the third denomination selects 1 of the
four available suits. The singleton card may be any one of the
three denominations. Thus, the calculation becomes: COMBIN(13,
3) * (COMBIN(4, 2))^2 * COMBIN(4, 1) * 3 = 123,552 possible
hands with 2 pairs.
Three of a kind is calculated in a similar
manner. There will be 3 different denominations from the 13
possible denominations. One denomination will select 3 of the
4 available suits while the other two denominations select 1
card from each of the 4 possible suits. Finally, the three of
a kind can be in any of the three denominations. The
calculation becomes: COMBIN(13, 3) * COMBIN(4, 3) * (COMBIN(4,
1))^2 * 3 = 54,912 possible hands with 3 of a kind.
The next calculation will be for a Full House. A
Full House only uses 2 of the 13 denominations. One of these
will select 3 cards from the 4 available while the other
selects 2 cards from the 4 available. Finally the denomination
that has 3 cards can be either one of the 2 denominations that
we are using. This gives us: COMBIN(13, 2) * COMBIN(4, 3) *
COMBIN(4 , 2) * 2 = 3,744 possible Full Houses.
The final calculation is for 4 of a kind. Again,
we will select 2 denominations from the 13 available. One of
these will select 4 cards from the 4 available (Obviously the
only way to do this is to take all four cards.) while the
other denomination takes 1 of the available 4 cards. The
denomination that has 4 of a kind can be either one of the 2
available denominations. Thus, the calculation becomes:
COMBIN(13, 2) * COMBIN( 4, 4) * COMBIN( 4, 1) * 2 = 624
different ways of being dealt 4 of a kind. (On the draw, ask
one of the other players what the odds are of drawing to an
inside straight. Then draw your card. It won't make any
difference though as no one else will have anything, and they
will all fold.)
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