What’s the probability of being dealt a full house in 5-card poker?

Answer: The probability of being dealt a full house in 5-card poker is approximately 0.00144 or 0.144%.

1. Understanding a Full House: In the game of poker, a full house is a strong hand consisting of three cards of one rank and two cards of another rank. For example, you might have three Kings and two Fives.

2. Total Number of 5-Card Hands: In a standard deck of 52 playing cards, many possible combinations of 5-card hands can be dealt. To calculate this, we use the concept of combinations. The number of ways to choose 5 cards from a deck of 52 is calculated as “52 choose 5”, which is written as . This value is equal to 2,598,960.

3. Counting Full House Hands: To find the number of full house hands, we need to consider how many ways we can choose the ranks for the three-of-a-kind and the pair. There are 13 ranks in a standard deck (Ace through King). Once we’ve chosen the ranks, we need to determine how many ways we can select the actual cards for each rank. For the three-of-a-kind, we have 4 cards of each rank (one in each suit), so there are 4 ways to choose each card. For the pair, we have 4 cards of each rank remaining except for the rank already chosen for the three-of-a-kind, so there are 4-1=3 ways to choose each card. So, the total number of full house hands is:

• Choose the rank for the three-of-a-kind: There are 13 possible ranks to choose from.
• Choose the rank for the pair: Once we’ve chosen the rank for the three-of-a-kind, we have 12 remaining ranks to choose from for the pair.

Number of ways to choose ranks×Number of ways to choose cards for three-of-a-kind×Number of ways to choose cards for the pair Number of ways to choose ranks×Number of ways to choose cards for three-of-a-kind×Number of ways to choose cards for the pair

=13×4×12×3=13×4×12×3=1872=1872

4. Calculating Probability: The probability of being dealt a full house is the ratio of the number of full house hands to the total number of possible 5-card hands:Probability=Number of full house handsTotal number of 5-card hands:

5. Express as Percentage: To express this probability as a percentage, we multiply by 100:

Probability≈0.000720×100Probability≈0.000720×100≈0.144%≈0.144%

So, the probability of being dealt a full house in 5-card poker is approximately 0.144%. This means that in a randomly dealt 5-card hand, there’s about a 0.144% chance that you’ll have a full house.