Poker Probabilities with Seven Card Deal

Date: 03/19/2004 at 06:10:33
From: Paul
Subject: Poker Probabilites

As I looked through the posted answers I note that there is nothing I 
could find if you were dealt seven cards vs. five cards, as in Seven 
Card Stud or Texas Hold-Em.  With the added two cards being dealt how
does that effect the probabilities, or does it?

Date: 03/19/2004 at 16:04:58
From: Doctor Roy
Subject: Re: Poker Prbabilites

Hi Paul,

Thanks for writing to Dr. Math.

The probabilities are greatly affected by using 7 cards instead of 5.
You have a greater than 80% chance of having at least a pair.  Here's
the breakdown of the hands.

There are C(52,7) total ways of getting 7 cards out of a standard 
deck.  That's 133,784,560 possible 7 card hands.

Now, let's go through the number of ways to get each particular hand.

Royal Flush:  With five cards, there are only 4 possible royal flushes
(one of each suit).  With 7 cards, we have to consider the 6th and 7th
cards.  After the 5 cards for the royal flush, there are 47 cards left
over.  There are C(47,2) = 1081 ways of picking the last two cards, or
4324 total Royal flushes possible.

Prob(Royal flush) = 4*1081 / C(52,7) = 0.00323%

Straight Flush:  There are 9 possible straights possible for a
straight flush that is not a royal flush, one each for the lowest card
in the straight being anything from ace to 9.  There are 4 suits.  The
last two cards can be any card except the card making a higher
straight flush.  For example, if you had A-2-3-4-5 of spades, you
don't want to let one of the other cards be a 6 of spades, or you will
be counting that possibility twice.  That leaves 46 cards for the last
two, and C(46,2) = 1035.  9*4*1035 = 37260 straight flushes possible.

Prob(Straight flush) = 4*9*C(46,2) / C(52,7) = 0.0279%

Four of a kind:  There are 13 different unique 4-of a kinds (one for
each rank).  There are C(48,3) ways of picking the remaining three
cards.  So, there are 13*C(48,3) = 224,848 possible Four of a Kinds.

Prob(Four of a kind) = 13*C(48,3) / C(52,7) = 0.168%

Full Houses:  This gets a bit complicated.  There are 3 separate cases
we must consider.

Case 1:  We could have two sets of triples and an extra card.  There
are C(13,2) ways of picking which two ranks are the triples.  For each
triple, there are C(4,3) ways of picking which 3 of the 4 cards of a
rank are in the triple.  Finally, there are 44 extra cards to be used
as the 7th card.  That's a total of C(13,2)*C(4,3)*C(4,3)*44 = 54,912
ways of getting this type of full house.

Case 2:  1 set of triples, 2 different unique pairs.  There are 13
ways to pick the rank of the triple (any of one the ranks).  There are
C(12,2) ways to pick the rank of the pairs.  For the triples, there
are C(4,3) ways of picking the suits.  For each pair, there are C(4,2)
ways of picking the suits of the pair.  That's a total of
13*C(12,2)*C(4,3)*C(4,2)*C(4,2) = 123,552 ways of getting this type of
full house.

Case 3:  1 set of triples, 1 pair, 2 blanks (don't match either the
triple, the pair, or each other).  There are 13 ways to pick the rank
of the triples.  There are 12 ways to pick the rank of the pair.  And
there are C(11,2) ways to pick the ranks of the blanks.  There are
C(4,3) ways of picking the suits of the triples.  There are C(4,2)
ways of picking the suits of the pair.  And for each blank, there are
4 suits available.  So, there's a total of
13*12*C(11,2)*C(4,3)*C(4,2)*4*4 = 3,294,720 ways of getting this type
of full house.

In all, there are 3,473,184 ways of getting a full house.

Prob(Full house) = 3,473,184 / C(52,7) = 2.60%
 
Flush:  For a flush, we again have 3 cases.  We'll take each one at a
time.

Case 1:  All 7 cards in one suit.  There are C(13,7) ways of picking 7
different cards in one suit.  But we want to get rid of possibilities
for straights.  There are 8 ways all 7 cards could form a 7-card
straight.  There are 47 ways to form 6 card straights, and there are
162 ways to form 5 card straights from the 7 cards.  That's a total of
217 ways.  So, there are 4*(C(13,7) - 217) = 5996 possible flushes of
this type.

Case 2:  6 cards of one suit, and a blank.  There are C(13,6) ways of
picking 6 different cards in one suit.  There are 9 ways to get a 6
card straight here and 62 ways of getting a 5 card straight.  And 39
ways the blank could come up.  That's a total of 4*(C(13,6) - 71)*39 =
256,620 possible flushes of this type.

Case 3:  5 cards of one suit, and two blanks.  There are C(13,5) ways
of picking 5 different cards in one suit.  We remove the 10 possible
straights to get C(13,5) - 10.  The last two cards cannot give us a
hand better than a flush.  So, we compute the probability directly.
C(39,2) ways of picking the last two cards.  4*(C(13,5) - 10)*C(39,2)
= 3,785,028 possible flushes of this type.

Adding them together, there are 4,043,147 ways of getting a flush.

Prob(Flush) = 4043147/C(52,7) = 3.02%

Straights:  We calculate this in multiple parts as well.

Case 1:  7 distinct cards (no paired cards).  Above, we calculated 217
ways to get a 5 card straight from 7 cards.  All we have to do now is
discount flushes.  There are 4 ways to choose all 7 cards in any of
the 4 suits.  For 6 cards in the same suit, there are 4 suits, and
C(7,6) ways of picking the 6 cards, and 3 choices for the suit of the
remaining card, or 4*C(7,6)*3 = 84 ways of a flush in this situation.
For 5 cards in the same suit, there are 4 suits, C(7,5) ways of
picking the 5 cards, and 3 choices for suit for each of the remaining
two cards, for a total of 4*C(7,5)*3*3 = 756.  We subtract 756, 4, and
84 from 4^7, the total number of ways the cards can be suited, 4^7 -
756 - 4 - 84 = 15540.  So, with 7 distinctly ranked cards with a
straight but without a flush, there are 217*15540 = 3,372,180 ways of
making it happen.

Case 2:  6 distinct cards (last card pairs one of the others).  There
are 9 ways to have 6 card straights.  We can also have a 5 card
straight with the 6th distinct rank separate.  If the straight begins
with A or 10, there are 7 choices for the 6th card.  Otherwise, there
are 6 choices.  That's a total of 2*7 + 6*8 = 62 ways.  In all, there
are 71 ways of producing a 5 card straight from 6 distinct cards.

There are 6 choices for which distinct card is paired.  There are
C(4,2) = 6 ways to pick which suits make up the pair.  The remaining
cards can be chosen from any suit, so there are 4^5 ways to pick the
suiting.  We now get rid of flushes.  There are 4 ways to pick all the
5 non-paired cards all in the same suit.  In addition, we can't have 4
of them match the suit of one of the paired cards.  There are C(5,4)
ways of picking which 4, 2 choices for which paired card is picked for
suit, and 3 choices for the suit of the 5th card (that is a different
suit) for a total of 4 + C(5,4)*2*3 = 34.  So, we subtract this from
the total ways to suit the 5 cards: 4^5 - 34 = 990.  That makes for
71*36*990 = 2,530,440 ways of a straight of this type. 

Case 3:  5 distinct cards (must include two pairs or a three of a
kind).  There are 10 ways to make a straight with 5 distinct cards.
First, let's consider the three of a kind.  There are 5 different
possible three of a kinds.  There are C(5,4) for which suits comprise
the trips.  The suits of the remaining 4 cards can be any of the 4
suits, for a total of 4^4 = 256.  But we should eliminate the three
possibilities that match the suits with any in the trips (to prevent
flushes).  So, with trips, that's a total of 10*5*4*(256-3) = 50,600.

For the case with two pairs, there are C(5,2) ways of picking which
two cards are the two pair.  For each pair, there are C(4,2) = 6 ways
of picking suits.  6*6 = 36.  But now, we have to split the 36 up into
cases to get rid of flushes.

6 of them have each pair matching two suits.  24 of them have one suit
matched, and 6 of them have no matching suits.

There are 4^3 = 64 ways of suiting the three remaining cards in the
hand.  Of these, if two suits are matched, we don't want either
possibility of flush to come up.  So, in this case, we have 62 ways of
suiting the remaining cards.  If only one suit matches in the two
pairs, there's only 1 way of matching the suit, and 63 safe ways of
suiting the remaining cards.  If both two pairs have different suits,
all 64 ways of suiting are safe.  So, for a 5 card straight with trips
or with two pair, that's a total of: 50600 + 10*C(5,2)*[6*62 + 24*63 +
6*64] = 277,400 ways of getting a straight. 

Adding all 3 cases together, we get 6,180,020 ways of getting a straight.

Prob(Straight) = 6,180,020 / C(52,7) = 4.62%

Three of a kind:  A hand that is three of a kind must have 5 distinct
ranks of cards (fewer and you have a full house or two pair or
something else).  There are C(13,5) ways of picking the ranks, of
which we remove the 10 possible straights.  That leaves 1277
possibilities.  We can pick any of the 5 ranks for the trips, and
there are C(5,4) ways of suiting the trips.  Of the remaining 4 cards,
we want to get rid of the possibility of a flush.  There are 4^4 = 256
ways of suiting the 4 cards and 3 ways of completing the flush.  So,
there are 253 safe ways of completing the flush.

That's a total of 1277*5*C(5,4)*253 = 6,461,620 ways of getting a
three of a kind.

Prob(Three of a kind) = 6,461,620 / C(52,7) = 4.83%

Two pair: 

Case 1:  As above, a hand that is two pair can have 5 distinct ranks,
leaving 1277 qualifying hands 5 rank hands.  As in the case above for
straights, there are 10 ways of picking which two pair and [6*62 +
24*63 + 6*64] ways of not also making a flush.  That's a total of:
1277*10*[6*62 + 24*63 + 6*64] = 28,962,360 ways of getting two pair of
this type.

Case 2:  Three pairs plus a blank card.  There are C(13,3) ways of
picking the rank of the cards.  For each pair, there are C(4,2) ways
of picking the suits, and 40 cards left over for the blank.  That's a
total of C(13,3)*C(4,2)^3 * 40 = 2,471,040 ways of getting two pair of
this type.

In all, there are 31,433,400 ways of getting two pair.

Prob(Two Pair) = 31,433,400 / C(52,7) = 23.50%

Pair:  In a pair, there must be 6 distinct ranks in the hand.  Above,
we saw there were C(13,6) - 71 = 1645 ways of doing this without
straights showing up.  There are 6 choices for which rank is paired,
and C(4,2) ways of picking the suits of the pair.  The suits of the
remaining cards can be any of the 4, leaving 4^5 choices.  We want to
eliminate flushes now.  We can't have all 5 cards in the same suit.
This can occur in 4 ways.  We also cannot have 4 of them match suits
with either card in the pair.  There are C(5,4) ways of choosing which
4 cards, 3 ways of picking the suit of the extra card, and 2 suits to
choose from in the pair, or C(5,4)*2*3 = 30.  So, of the 4^5 ways of
suiting the 5 cards, 4^5 - 30 - 4 = 990 are safe.  In total, there are
1645*6*6*990 = 58,627,800 ways of getting a pair.

Prob(Pair) = 58,627,800 / C(52,7) = 43.82%

High card:  For a high card situation, there must be 7 distinct ranks.
There can be no pairs.  Getting rid of straights, as above, we are
left with 1499 sets of 7 cards without a straight.  Now, we have to
get rid of flushes.  There are 4 ways for all 7 cards to have the same
suit.  For 6-flushes, there are C(7,6) ways of picking the 6 cards, 4
possible suits, and 3 possible suits for the last card.  For
6-flushes, C(7,6)*4*3 = 84.  For 5-flushes, there are C(7,5) ways of
picking the 5 cards, 4 possible suits, and 3 choices for each of the
remaining two cards for C(7,5)*4*3*3 = 756 ways to get a 5-flush.  Of
the 4^7 possible suitings of the 7 cards, 756 + 84 + 4 = 844 are bad.
So, there are a total of 1499*(4^7 - 844) = 23,294,460 possible ways
to get a high card.

Prob(High card) = 23,294,460 / C(52,7) = 17.41%

If we add up all the possibilities we calculated for each type of
hand, we get 133,784,560, which is the appropriate number, giving us
an indication our calculations are correct.

Here's a chart summarizing the information:

  Hand                Possibilities        Probability
  ---------------     -------------        -----------
  Royal Flush              4,234              0.00324%
  Straight Flush          37,260              0.0279%
  4 of a kind            224,848              0.168%
  Full House           3,473,184              2.60%
  Flush                4,043,147              3.02%
  Straight             6,180,020              4.62%
  Three of a Kind      6,461,620              4.83%
  Two Pair            31,433,400             23.50%
  Pair                58,627,800             43.82%
  High Card           23,294,460             17.41%

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/