Four Of A Kind Poker

In the standard game of poker, each player gets5 cards and places a bet, hoping his cards are 'better'than the other players' hands.

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Four Of A Kind Poker Meaning

Four-of-a-kind stands as the No. 3 hand in the standard poker hand rankings, only trailing the straight flush and the royal flush. A 52-card deck yields 156 distinct ways to make four-of-a-kind. Combined with all possible different kickers, there are 624 possible ways to draw four-of-a-kind.
Four of a kind, also known as quads, is a five-card poker hand in which four of the cards are of the same rank. For example, a player holds every Queen in the deck, or a player has three tens and a wild card.

'With the four cards each player is dealt, he or she must use two of them to make the best hand possible with three of the community cards. The use of two cards from the hand is a requirement. Also known as “quads,” this hand is another “monster” and will win the pot 99.9999 percent of the time. Example: Four of a kind Summary of the hand rankings. Contributed by William Moore of NY state, who says that it was 'introduced' in a 'family poker night' in the late-1950's (maybe '57 or '58), and became a rather popular addition to the games played. He taught to college friends in the late 1960's, but the regular Draw & Stud Poker games remained more popular with them.

The game is played with a pack containing 52 cards in 4 suits, consisting of:

13 hearts:
13 diamonds
13 clubs:
13 spades:

♥ 2 3 4 5 6 7 8 9 10 J Q K A
♦ 2 3 4 5 6 7 8 9 10 J Q K A
♣ 2 3 4 5 6 7 8 9 10 J Q K A
♠ 2 3 4 5 6 7 8 9 10 J Q K A

The number of different possible poker hands is found by counting the number of ways that 5 cards can be selected from 52 cards, where the order is not important. It is a combination, so we use `C_r^n`.

The number of possible poker hands

`=C_5^52=(52!)/(5!xx47!)=2,598,960`.

Royal Flush

The best hand (because of the low probability that it will occur) is the royal flush, which consists of 10, J, Q, K, A of the same suit. There are only 4 ways of getting such a hand (because there are 4 suits), so the probability of being dealt a royal flush is

`4/(2,598,960)=0.000 001 539`

Straight Flush

The next most valuable type of hand is a straight flush, which is 5 cards in order, all of the same suit.

For example, 2♣, 3♣, 4♣, 5♣, 6♣ is a straight flush.

For each suit there are 10 such straights (the one starting with Ace, the one starting with 2, the one starting with 3, ... through to the one starting at 10) and there are 4 suits, so there are 40 possible straight flushes.

The probability of being dealt a straight flush is

`40/(2,598,960)=0.000 015 39`

[Note: There is some overlap here since the straight flush starting at 10 is the same as the royal flush. So strictly there are 36 straight flushes (4 × 9) if we don't count the royal flush. The probability of getting a straight flush then is 36/2,598,960 = 0.00001385.]

The table below lists the number ofpossible ways that different types of hands can arise and theirprobability of occurrence.

Ranking, Frequency and Probability of Poker Hands

Hand	No. of Ways	Probability	Description
Royal Flush	4	0.000002	Ten, J, Q, K, A of one suit.
Straight Flush	36	0.000015	A straight is 5 cards in order. (Excludes royal and straight flushes.) An example of a straight flush is: 5, 6, 7, 8, 9, all spades.
Four of a Kind	624	0.000240	Example: 4 kings and any other card.
Full House	3,744	0.001441	3 cards of one denominator and 2 cards of another. For example, 3 aces and 2 kings is a full house.
Flush	5,108	0.001965	All 5 cards are from the same suit. (Excludes royal and straight flushes) For example, 2, 4, 5, 9, J (all hearts) is a flush.
Straight	10,200	0.003925	The 5 cards are in order. (Excludes royal flush and straight flush) For example, 3, 4, 5, 6, 7 (any suit) is a straight.
Three of a Kind	54,912	0.021129	Example: A hand with 3 aces, one J and one Q.
Two Pairs	123,552	0.047539	Example: 3, 3, Q, Q, 5
One Pair	1,098,240	0.422569	Example: 10, 10, 4, 6, K
Nothing	1,302,540	0.501177	Example: 3, 6, 8, 9, K (at least two different suits)

Question

The probability for a full house is given above as 0.001441. Where does this come from?

Answer

Explanation 1:

Probability of 3 cards having the same denomination: `4/52 xx 3/51 xx 2/50 xx 13 = 1/425`.

(There are 13 ways we can get 3 of a kind).

The probability that the next 2 cards are a pair: `4/49 xx 3/48 xx 12 = 3/49`

(There are 12 ways we can get a pair, once we have already got our 3 of a kind).

The number of ways of getting a particular sequence of 5 cards where there are 3 of one kind and 2 of another kind is:

`(5!)/(3!xx2!)=10`

So the probability of a full house is

`1/425 xx 3/49 xx 10 ` `= 6/(4,165)` `=0.001 440 6`

Explanation 2:

Number of ways of getting a full house:

`(C(13,1)xxC(4,3))` `xx(C(12,1)xxC(4,2))`