The Probability of Getting a Four-of-a-Kind Poker Hand

Because of its rareness, a four-of-a-kind card combination is one of the most aspired and dreamt about hand in poker. It is essentially made up of four cards all having the same value and an individual card. Together they form one of the most potent hands in the game. This hand is almost unbeatable by almost all the other card combinations in the poker hand rankings except for a couple of very special hands.

A four-of-a-kind can only be beaten by two special card combinations. The straight flush and the royal flush are the only hands that can put away this pretty potent hand. A straight flush is basically made up of five cards under a single suit. They can also be arranged in sequential order. Meanwhile, a royal flush is just like a straight flush. The only difference is that the royal flush is made up of all face cards.

Aside from these two, all the other poker hands succumb to the strength and power of a four-of-a-kind. This hand is almost unbeatable when played on any situation. The no pair/high card, one pair, and two pair card combinations are simply helpless against it. The three-of-a-kind, straight, and flush are just as powerless over it. Even a full house is limited against the strength of a four-of-a-kind.

According to some mathematical calculations, the frequency of getting this hand is very low already. The total number of card combinations that could contain a four-of-a-kind is 624. Because of this ultra low frequency, the probability of getting such a strong hand falls as well. It is measured at the contemptibly small rate of 0.024 percent.

Upon analyzing the probabilities of each and every poker hand, it can be observed that the probability of getting these card combinations decreases as their value increase. Simultaneously, the probability of getting these poker hands increases as their values submerge. These are dominant trends that are usually observed when checking and analyzing the different probabilities of each and every hand in poker.

The opposite relationship between poker hand value and poker probability and frequency is proven by precise mathematical calculations. It means that a four-of-a-kind, which is generally considered a strong card combination, has little probability and frequency because of its relatively high value. On the contrary, low poker hands such as the no pair/high card, two pair, and three-of-a-kind have higher probability and frequency due to their

relatively low value.

Poker 4 of a kind probability

The Probability of Four of a Kind (fraction) constant defines the probability of being dealt four-of-a-kind and is represented as a fraction. The Four of a Kind hand is a five card hand having four of five cards being the same value cards. For example:

  • 4 of spades
  • 4 of hearts
  • 4 of diamonds
  • 4 of clubs, and
  • an 8 of spades

would be a Four of a Kind hand.

The number of possibilities of a Four of a Kind hand is the number of card values , `((13),(1))` ; times the number of possible card values of the fifth card (the card that is not part of the four off a kind) , `((12),(1))` ; times the number of suits of that fifth card could be , `((4),(1))` . The total number of five card hands possible is computed as `((52),(5))` (see the constant defining number of five card hands ), so this constant represent the ratio of number of possible Four of a Kind hands to the number of five card hands possible, `((52),(5))` .

Additional Resources:

The Poker Calculator provides all the equations, the calculations, to determine the probabilities of success and failure when playing Poker with a standard five card deck.

[Card Games | Poker | Combinatorics | Probability | Chance]

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