What is the probability of finding exactly two pairs in a five-card hand?
Answer (1)
To find the probability of getting exactly two pairs in a five-card poker hand, we need to follow these steps:
Understand the Composition of Two Pairs: A hand with two pairs consists of:
Two cards of one rank.
Two cards of another rank.
One card of a different rank (kicker).
Calculate the Total Number of Possible Hands: A standard deck of cards has 52 cards. The number of possible 5-card hands is given by the combination formula: ( 5 52 ) = 5 × 4 × 3 × 2 × 1 52 × 51 × 50 × 49 × 48 = 2 , 598 , 960
Calculate the Number of Ways to Get Two Pairs:
Choose 2 ranks for the pairs. There are 13 ranks available (2 to Ace), so: ( 2 13 ) = 78
For each rank chosen, choose 2 suits (since each rank has 4 suit options). The number of ways to choose 2 suits from 4 is: ( 2 4 ) = 6
Therefore, for two ranks there are: 6 × 6 = 36
Choose a different rank for the fifth card (the kicker). There are 11 remaining ranks after choosing two for the pairs, so: ( 1 11 ) = 11
Choose the suit for the kicker. There are 4 options: ( 1 4 ) = 4
Multiply these to get the total number of ways to arrange a two pair hand: 78 × 36 × 11 × 4 = 123 , 552
Calculate the Probability: Finally, divide the number of favorable outcomes by the total number of possible hands: 2 , 598 , 960 123 , 552 ≈ 0.0475
So, the probability of being dealt exactly two pairs in a five-card hand is approximately 4.75%.
By analyzing and calculating using combinations, you can find the precise probability desired for this poker hand scenario.
