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Probability and Poker
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Odds on One Card

It is well to note that the possibility of getting any one particular card to make up a hand is necessarily one in 52 minus the known cards. In draw poker that one card is going to have 1 chance in 47 of showing up. This means that of every 47 times you try for the card, in all likelihood, it will only turn up once. If you are looking for two specific cards, your odds are tremendously longer for you will have to multiply the odds for getting each card together. Thus the odds to get two particular cards is 2 in 47 multiplied by 1 in 46, which equals 1 chance in 1081. Your odds get longer as you look for more cards.

As you go through the various hands keep this in mind. If you can settle for any one of several cards or there are wild cards in the deck the dynamic will change. Each wild card in a deck drastically increases the odds of getting a particular card (by an arithmetic progression equivalent to the number of wild cards). Thus, a player looking for a particular card will now have 3 chances in 47 if jokers are wild or 5 chances in 47 if deuces are wild.

So even if the odds are very long that you will be dealt a particular hand, once the hunt comes down to one particular card, your chances of filling out your hand compared with another player trying to fill out a hand with only one unique card are exactly even.

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