Five cards are dealt from a well-shuffled standard deck. using combinations, write an expresssion for the probability all are hearts
To find the probability that all five cards are hearts when dealing from a well-shuffled standard deck, we need to determine the number of ways to select five cards from the 13 hearts in the deck divided by the total number of ways to select five cards from the 52 cards in the deck.
Let's break down the calculation step by step:
1. Calculate the number of ways to select five cards from the 13 hearts:
This is equivalent to finding the number of combinations of 13 objects taken 5 at a time, which can be denoted as C(13,5) or "13 choose 5." The formula for calculating combinations is given as:
C(n, k) = n! / (k! * (n-k)!), where n is the total number of objects, and k is the number of objects to be selected.
Plugging in the values, we have:
C(13, 5) = 13! / (5! * (13-5)!)
2. Calculate the total number of ways to select five cards from the 52 cards in the deck:
This is equivalent to finding the number of combinations of 52 objects taken 5 at a time, denoted as C(52,5). Following the same formula:
C(52, 5) = 52! / (5! * (52-5)!)
3. Finally, to find the probability, we divide the number of ways to select five hearts by the total number of ways to select five cards:
P(all five hearts) = C(13, 5) / C(52, 5)
Hence, the expression for the probability that all five cards are hearts is:
P(all five hearts) = C(13, 5) / C(52, 5)