Final answer:
The probability of being dealt two cards with the same denomination and three cards with other denominations in poker is 0.5890.
Step-by-step explanation:
In poker, you are dealt 5 cards at random from a standard deck of 52 cards. To calculate the probability of being dealt two cards with the same denomination (pair) and three cards with other denominations, we need to consider the number of possible pairs and the number of possible combinations for the remaining three cards.
Step 1: Determine the number of possible pairs. There are 13 denominations in a deck (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). For each denomination, there are 4 cards (one in each suit). Therefore, there are 13 * (4 choose 2) = 78 possible pairs.
Step 2: Determine the number of possible combinations for the remaining three cards. After removing the two cards of the pair, there are 50 cards remaining in the deck. We need to choose 3 cards from these 50. Therefore, the number of possible combinations is (50 choose 3) = 19600.
Step 3: Calculate the probability. The probability of getting two cards with the same denomination and three cards with other denominations is the ratio of the number of favorable outcomes to the number of possible outcomes. The number of favorable outcomes is 78 (pairs) * 19600 (combinations) = 1528800. The number of possible outcomes is (52 choose 5) = 2598960. Therefore, the probability is 1528800 / 2598960 = 0.5890 (rounded to 4 decimal places).