# Problem 62 Solution - Probabilities in Poker

## Probabilities of drawing

Below are the probabilities of drawing the following hands, without discarding:

```
Approximate
Hand             Exact Probability     Probability
---------------   ----------------------   ------------
Royal Flush:              4 in 2,598,960   1 in 649,740
Straight Flush:          36 in 2,598,960   1 in 72,193
Four of a Kind:         624 in 2,598,960   1 in 4,165
Full House:           3,744 in 2,598,960   1 in 694
Flush:                5,108 in 2,598,960   1 in 509
Straight:            10,200 in 2,598,960   1 in 255
Three of a Kind:     54,912 in 2,598,960   1 in 47
Two Pair:           123,552 in 2,598,960   1 in 21
One Pair:         1,098,240 in 2,598,960   1 in 2.4
Nothing:          1,302,540 in 2,598,960   1 in 2.0

```

## Derivations

If you are not familiar with the factorial function, the factorial of a number is the product of every integer from one to itself. For example five factorial, denoted 5!, is 1*2*3*4*5 = 120.

If you had x cards and chose y of them, the number of unique sets you could create would be x!/(y!*(x-y)!). For the purposes of this document I shall notate this is (x:y).

The number of ways to arrange 5 cards out of 52 is (52:5) = 2,598,960. The odds of drawing any given hand are the number of ways it can be arranged divided by the total number of ways to arrange five cards above. Below are the number of ways to arrange each hand.

Royal Flush

The number of different royal flushes are four (one for each suit).

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and (4:2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.

Flush

There are 4 suits to choose from and (13:5) = 1,287 ways to arrange five cards in the same suit. Then subtract the royal and straight flushes to avoid double counting. The total number of flushes is 4 * 1,287 - 4 - 36 = 5,108.

Straight

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. Then subtract the royal and straight flushes to avoid double counting. Thus the number of ways to arrange a straight is 10 * 45 - 4 - 36 = 10,200.

Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are (12:2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 42 = 54,912.

Two Pair

There are combin(13,2)=78 ways to choose 2 ranks out of 13 for the two pair. Then there are 11 ranks left for the singleton. For each pair there are combin(4,2)=6 ways to choose 2 suits out of 4. For the singleton there are 4 possible suits to chose from. So the total two pair combinations are 78*11*6*6*4 = 123,552.

One Pair

There are 13 ranks to choose from for the pair and (4:2) = 6 ways to arrange the two cards in the pair. There are (12:3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 43 = 1,098,240 ways to arrange a pair.

Nothing

There must be five different ranks represented, of which there are (13:5) = 1,287 possible combinations. Each rank has four cards to choose from. Finally subtract the number of straights, flushes, straight flushes, and royal flushes, to avoid double counting. Thus the number of ways to arrange nothing is 1,287 * 45 - 4 - 36 - 5,108 - 10,200 = 1,302,540.