91Ó°ÊÓ

For a full house, we need to select a rank for the three-of-a-kind and a different rank for the pair. There are 13 ranks in a deck of cards (from Ace to King). We will first choose the rank for three-of-a-kind and then choose the rank for the pair. - To choose the rank for three-of-a-kind (3 cards of the same rank): There are 13 ranks and we need to select 1, so there are \( _{13}C_{1} \) ways to do this. - To choose the rank for the pair (2 cards of another rank): After choosing the rank for three-of-a-kind, we have only 12 ranks left, and we need to choose 1, so there are \( _{12}C_{1} \) ways to do this. Now that we have chosen the ranks, we need to choose the suits for each rank. - For the three-of-a-kind: There are 4 cards available for each rank (one for each suit), and we need to choose 3 of them. So, there are \( _{4}C_{3} \) ways to do this. - For the pair: There are 4 cards available for this rank, and we need to choose 2 of them. So, there are \( _{4}C_{2} \) ways to do this. To find the total number of full house hands, we multiply the number of ways to select the ranks and the suits: Total full house hands = \((_{13}C_{1})\cdot (_{12}C_{1})\cdot (_{4}C_{3})\cdot (_{4}C_{2})\)

To find the total possible 5-card poker hands, we use the combination formula, since the order of the cards doesn't matter. Total possible poker hands = \(_{52}C_{5}\)

Now that we have both the total number of full house hands and the total possible 5-card poker hands, we can calculate the probability of getting a full house: Probability of a full house = \(\frac{\text{Total full house hands}}{\text{Total possible poker hands}}\) Probability of a full house = \(\frac{(_{13}C_{1})\cdot (_{12}C_{1})\cdot (_{4}C_{3})\cdot (_{4}C_{2})}{_{52}C_{5}}\) After plugging in the values for the combinations, simplify the expression: Probability of a full house = \(\frac{(13)\cdot (12)\cdot (4)\cdot (6)}{2,598,960}\) Probability of a full house = \(\frac{3,744}{2,598,960}\) Simplify the fraction: Probability of a full house ≈ \(0.001440576\) Thus, the probability of being dealt a full house in a 5-card poker hand is approximately 0.144%.