Final answer:
To find the probability of a three of a kind in a 5-card poker hand, we need to calculate the total number of possible hands and the number of three of a kind hands.
Step-by-step explanation:
To find the probability of obtaining a three of a kind in a 5-card poker hand, we need to consider the total number of possible poker hands and the number of three of a kind hands.
First, let's determine the total number of possible 5-card poker hands. There are 52 cards in a deck, and we want to choose 5 of them, so we use the combination formula:
C(52, 5) = 52! / (5! * (52-5)!)
Simplifying, we get:
C(52, 5) = 2,598,960
Now we need to determine the number of three of a kind hands. The three of a kind hand consists of three cards of the same rank and two other cards of different ranks.
There are 13 ranks in a deck, so we choose one of them for the three of a kind and two different ranks for the other two cards. The number of three of a kind hands can be calculated as:
13 * C(4, 3) * C(12, 2)
Simplifying, we get:
13 * 4 * 66 = 3,744
Finally, we can find the probability by dividing the number of three of a kind hands by the total number of possible hands:
Probability = 3,744 / 2,598,960 = 0.00144