Probabilities of Getting Dealt a Particular Poker Hand

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Probabilities of Getting Dealt a Particular Poker
Hand
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• First we must compute the total number of
  possible 5-card combinations. The number of ways
  5 cards can be drawn from a deck of 52 would be
  52C5 or (52!)/(52-5)!(5!) 52!/47!5!
  2,598,960 possible hands.

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• A straight is a combination of 5 consecutive
  numbers of any suit (excluding all being the same
  suit). A straight would be 1) 2-3-4-5-6, 2)
  3-4-5-6-7, etc. The are 10 such ordered sets and
  each of the five cards can have any suit. Since
  we want straights excluding the possibility of
  having them in the same suit, we need to subtract
  the probability of getting a straight flush.

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• The probably of getting a straight flush is,
  again, 1 pulled from 10 sets, but this time all
  of them having a suit that is a single choice
  from 4 possibilities. So, 10C1 x 4C1 x 4C1 x 4C1
  x 4C1 - 10C1 x 4C1 (10)(4)(4)(4)(4)(4)
  (10)(4) 10,240 40 10,200 possible straight
  sets that are not straight flushes. The
  probability of getting a straight (excluding
  straight flushes) is 10,200/2,598,960 0.003924
  or 0.392.

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• If you haven't noticed already, it seems that the
  best strategy in many cases is to try and go for
  the pair of jacks or better. And in fact in most
  cases it is! Your odds of getting a dealt a Jacks
  or better hand is simply 16/48 x 15/47 10.6
  (assuming the first 3 cards dealt to you are not
  what you are looking for).
• Source http//www.pokercheatdevice.com/news/2520.
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