Final answer:
To calculate the probability of being dealt a hand with only red cards in poker, one must use the combination formula to select five red cards from the standard 26 red cards in a 52-card deck. The resulting probability is approximately 2.53%.
Step-by-step explanation:
The subject of this question is mathematics, specifically combinatorics and probability within the context of a card game. To find the probability of being dealt a hand with only red cards in poker, we need to recognize that a standard deck of 52 playing cards consists of 26 red cards (hearts and diamonds) and 26 black cards (spades and clubs).
To calculate the probability, we need to determine the number of ways to choose five red cards from the 26 available, without regard to order. This is given by the combination formula C(n, k) = n! / (k! * (n-k)!), where n is the total number of items to choose from, and k is the number of items to choose.
Using this formula, the number of combinations for selecting five red cards is C(26, 5). We then divide this by the total number of five-card combinations from the entire 52-card deck, which is C(52, 5), to get:
P(only red cards) = C(26, 5) / C(52, 5)
Calculating these values gives:
P(only red cards) = (26! / (5! * (26-5)!) / (52! / (5! * (52-5)!))
Which simplifies to:
P(only red cards) = 65780 / 2598960 ≈ 0.0253
So, the probability of being dealt a hand with only red cards is approximately 0.0253, or roughly 2.53%.