Question

Consider a poker game of 2 decks of 104 cards Compute / derive the number of ways for one pair, two pairs, three of a kind, straight, flush, full house, four of a kind, and straight flush (with royal flush a special kind of straight flush).

Answer 1

The number of possible 5-card hands is

¹⁰⁴C₅ = 91962520

1.) A royal flush

A royal flush is ace, king, queen, jack, and ten of the same suit. If we order the 5-card hand from highest card to lowest, the first card will be an ace.

There are four possible suits for the ace. But the trick is there are 2 decks.

So, ways to choose an ace from any deck = ⁴C₁ * ²C₁= 8

After that, the other four cards are completely determined. But all other card can be chosen in ²C₁ ways.

# royal flushes = 4 * 2 * 2 * 2 * 2 = 64

Dividing by the number of possible hands gives the probability: €

P(royal flush) = 64 / 91962520 = 6.95 * 10^-7

2.) A straight flush(excluding Royal flush):

A straight-flush (excluding royal flush) is all cards the same suit and showing consecutive numbers (but not the highest 5 consecutive numbers).

If we order the 5-card hand from highest number to lowest, the first card may be one of the following: king, queen, jack, 10, 9, 8, 7, 6, or 5. (Note: the ace may be the card above a king or below a 2, but we would have a royal flush if it were the card above the king.) There are 9 possibilities. After the first card, whose suit we may choose in 8 ways(2 decks), the remaining cards are completely determined.

# straight flushes = 9 * ⁸C₁ = 9*8 = 72

Subtracting the number of royal flushes and dividing by the number of possible hands gives the probability:

P(straight - flush) = 72 / 91962520 = 7.82 * 10^-7

3.) Four of a kind

A four-of-a-kind is four cards showing the same number plus any other card.

If we order the 5-card hand with the four-of-a-kind first, we have ²⁶C₁ choices for the number showing on the first four cards.

The remaining card will be any of the 100 remaining cards

fours = ²⁶C₁ * 100= 2600

P(4 - of - a - kind) = 2600 / 91962520 = 2.82 * 10^-5

Similarly, you can calculate for others as well.

Please find the below 2 links as well which can help in understanding poker concept very well.

https://www.ece.utah.edu/eceCTools/Probability/Combinatorics/ProbCombEx15.pdf

http://www.math.hawaii.edu/~ramsey/Probability/PokerHands.html