Hypergeometric distribution

With Hold'em up probability examples


hypergeometric distribution:

h(x; n, a, N)

x
n
a
N


Result


hypergeometric distribution sum from x1 to x2 both included:

Sumx1 to x2 h(x; n, a, N)

x1 x2 (how many you need of the outs, e.g. if you need either 3, 4 or 5 outs, then x1=3 and x2=5)
n (turns left, for preflop the n is 5)
a (outs)
N (deck, for preflop the deck is 50)


Result


calculator

enter expression, e.g. 2+2
supported extra functions:
  • b(n,r) binomial function
  • h(x,n,a,N) hypergeometric distribution
  • hs(x1,x2,n,a,N) cumulative hypergeometric distribution



Result

Example of use

Lets say you get the following cards 6 and 8

What are the odds that you get a flush at the showdown (from preflop to the 5th card) on the table?
  • You need at least 3 of the outs to get a flush, then x1=3
  • There are 5 cards to be turned at the preflop state, n=5
  • You dont mind getting 6 or 7 suited in total, (you still get flush) so you include those odds too
    Then you set x2 = n. If you only want 3 suited on the table in total, then you set x2 to same as x1
  • Since you got two of the suited, there are 11 left in the deck, a=11
  • You have received two cards, then the deck size is now 50. N=50
Insert these numbers and you get the probability ~0.064
Only about 6% chance!
If you're aiming for a flush you shouldn't count on it, you should fold your cards


What are the odds that you get at least a pair at the showdown? (card value matches of your hand)
  • You need at least 1 of the outs to get a pair, then x1=1
  • There are 5 cards to be turned at the preflop state, n=5
  • You dont mind getting more than one pair, (you still get minimum a pair) so you include those odds too
    Then you set x2 = n. If you only want a pair in total, then you set x2 to same as x1
  • There are three 8 and three 6 left in the deck, you have 6 outs, a=6
  • You have received two cards, then the deck size is now 50. N=50
Insert these numbers and you get the probability ~0.49
About 50% change!
That's why it's so much better playing with high cards, since the odds are equal for all


What are the odds that you get a...

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