To calculate the probability of being dealt a 5-card hand that consists solely of hearts from a standard deck of 52 cards, we need to consider a few key elements.
A standard deck of 52 cards contains 4 suits: hearts, diamonds, clubs, and spades, with each suit containing 13 cards. Therefore, when we are dealing with hearts, we have a total of 13 cards to choose from.
The first step is to calculate the total number of ways to choose 5 cards from the total of 52 cards. This is represented mathematically by the combination formula:
Combination Formula: C(n, k) = n! / (k! * (n - k)!)
Where n is the total number of items to choose from, k is the number of items to choose, and ! represents factorial.
So, the total combinations of 5 cards from 52 is:
C(52, 5) = 52! / (5! * (52 - 5)!) = 52! / (5! * 47!)This simplifies to:
C(52, 5) = 2,598,960Now, we need to determine how many ways we can choose 5 cards from the 13 hearts available. Using the same combination formula, we find:
C(13, 5) = 13! / (5! * (13 - 5)!) = 13! / (5! * 8!)This simplifies to:
C(13, 5) = 1,287Now, to find the probability that all 5 cards dealt are hearts, we divide the number of successful outcomes (selecting 5 hearts) by the total number of possible outcomes (selecting any 5 cards):
Probability: P(all 5 are hearts) = C(13, 5) / C(52, 5)
Plugging in the values we calculated:
P(all 5 are hearts) = 1,287 / 2,598,960After performing this division:
P(all 5 are hearts) ≈ 0.0004952So, the probability of being dealt a 5-card hand consisting entirely of hearts from a standard 52-card deck is approximately 0.0004952, or about 0.04952%.