Question

1. How many 5 card poker hands are possible from a standard deck of cards?

2. What is the probability of a Royal Flush?

3. What is the probability of a Full House?

4. What is the probability of Two Pair?

5. How many more times likely is Two Pair than a Full House?   

Answer 1

1.) As we know there are 52 cards of 13 different types (each 4 ) in a standard deck of cards .

Possible number of 5 card poker hands = ^{= 2598960}

where    means choosing 5 out of 52.

2.) Royal flush consists of the ten, jack, queen, king, and ace of one suit. There are four suits.

So, the probability is = 4/2598960 = 0.000001539

3.) Full house has the pattern AAABB where A and B are from distinct kinds. The number of such hands is (13-choose-1)(4-choose-3)(12-choose-1)(4-choose-2).

Probability = (13x12x4x6) / 2598960 = 0.0014405762

4.) Two pair has the pattern AABBC where A, B, and C are from distinct kinds. The number of such hands is (13-choose-2)(4-choose-2)(4-choose-2)(11-choose-1)(4-choose-1).

After dividing by (52-choose-5), the probability is = 0.047539.

5.) The probability of two pair = 0.047539

The probability of full house = 0.0014405762

Two pair is = 0.047539 / 0.0014405762 = 32.9999 = almost 33 times more likely than a full house..