To calculate the probability of being dealt a five-card poker hand that includes the Ace of Hearts, we first need to consider the total number of possible five-card poker hands and the number of hands that contain the Ace of Hearts.
1. **Total Number of Five-Card Poker Hands:**
There are 52 cards in a standard deck. To find the total number of ways to choose 5 cards from 52, we use the combination formula:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of items to choose from (in this case, 52 cards), k is the number of items to choose (5 cards), and ! denotes factorial.
C(52, 5) = 52! / (5!(52-5)!) = 52! / (5!47!) = 2,598,960.
2. **Total Number of Hands Containing the Ace of Hearts:**
Now, we’ll determine how many of those hands include the Ace of Hearts. If we know one card (the Ace of Hearts) must be in the hand, we only need to choose 4 more cards from the remaining 51 cards in the deck.
C(51, 4) = 51! / (4!(51-4)!) = 51! / (4!47!) = 230,300.
3. **Calculating the Probability:**
Now that we have the number of successful outcomes (hands that contain the Ace of Hearts) and the total possible outcomes (all five-card hands), we can calculate the probability:
P(Ace of Hearts in hand) = Number of successful outcomes / Total outcomes
P(Ace of Hearts in hand) = C(51, 4) / C(52, 5) = 230,300 / 2,598,960.
This results in:
P(Ace of Hearts in hand) ≈ 0.0884, or about 8.84%.
In summary, the probability of being dealt a five-card poker hand that contains the Ace of Hearts is approximately 8.84%. This means that roughly 1 in every 11 poker hands will contain the Ace of Hearts. Understanding this concept not only adds to your strategic planning in poker but also enriches your overall appreciation of the game!