To find the probability that a five card poker hand contains cards of five different kinds, we must first understand what is meant by ‘five different kinds’. This means that we want one card from each of the five different ranks available in a standard deck of cards (which has 13 ranks: Ace, 2, 3, …, 10, Jack, Queen, King).
Here’s how to calculate it:
- Calculate the total number of five-card hands: The total number of ways to choose 5 cards from a 52-card deck is given by the combination formula:
- Calculate the number of favorable outcomes (hands with five different ranks): First, we need to choose 5 ranks from the 13 available ranks. The number of ways to do this is:
- For each of the 5 ranks chosen, we can choose one card from the 4 available suits: Since there are 4 suits (hearts, diamonds, clubs, and spades) for each rank, we can choose one card for each rank in:
C(52, 5) = 52! / (5!(52-5)!) = 2,598,960
C(13, 5) = 13! / (5!(13-5)!) = 1,287
4^5 = 1024
Now, to find the total number of hands with five different kinds, we multiply the number of ways to choose the ranks by the number of ways to choose the suits:
Total = C(13, 5) x 4^5 = 1,287 x 1,024 = 1,318,912
Finally, we can find the probability by dividing the number of favorable outcomes by the total number of possible five-card hands:
Probability = (Number of favorable outcomes) / (Total number of hands) = 1,318,912 / 2,598,960 ≈ 0.5086
This means that the probability of drawing a five card poker hand with cards of five different kinds is approximately 0.5086, or about 50.86%.