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Blaise Pascal also contributed to probability theory. Determined to know why his strategy was unsuccessful, he consulted with Pascal.
Pascal's work on this problem began an important correspondence between him and fellow mathematician Pierre de Fermat Communicating through letters, the two continued to exchange their ideas and thoughts.
These interactions led to the conception of basic probability theory. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling.
The following chart enumerates the absolute frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement.
Wild cards are not considered. In this chart:. The royal flush is a case of the straight flush. It can be formed 4 ways one for each suit , giving it a probability of 0.
The 4 missed straight flushes become flushes and the 1, missed straights become no pair. Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits.
So eliminating identical hands that ignore relative suit values, there are only , distinct hands. The number of distinct poker hands is even smaller.
However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q high card hand.
There are 7, distinct poker hands. In some popular variations of poker such as Texas Hold 'Em , a player uses the best five-card poker hand out of seven cards.
The frequencies are calculated in a manner similar to that shown for 5-card hands, except additional complications arise due to the extra two cards in the 7-card poker hand.
It is notable that the probability of a no-pair hand is less than the probability of a one-pair or two-pair hand. The Ace-high straight flush or royal flush is slightly more frequent than the lower straight flushes each because the remaining two cards can have any value; a King-high straight flush, for example, cannot have the Ace of its suit in the hand as that would make it ace-high instead.
Since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits.
Eliminating identical hands that ignore relative suit values leaves 6,, distinct 7-card hands. Implied odds changes things. But, if you expect your opponent to call a bet or raise on the river if you make your hand, your implied odds are or You'll often be asking this question if you're drawing to a straight or a flush.
So you'll need to calculate if you're getting good enough odds to call a bet or raise on the flop or turn. First, you need to calculate how often you'll hit your draw - by first counting your outs.
If you're drawing to a flush, you have two suited cards in your hand and two on the board, that means 9 cards of that suit left in the deck.
With pot odds, think of the number of cards again. That means 47 unseen cards including your opponents' hole cards. Nine cards can save you but 38 cards don't complete your draw.
This ratio changes again when you consider implied odds. So, say your opponent has a hand lesser than a flush, like two pair. Pot odds refers to the relationship between the size of the pot and the size of the bet.
Then you are getting pot odds of How big is the pot; how big is the bet? The more players involved in a pot, the less your chances of winning it.
That's why it may make sense to shove pre-flop with certain hands instead of just calling, hoping to narrow the field to just one, or perhaps zero!
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