To calculate the probability that a five-card poker hand dealt to you contains exactly 4 Aces, we need to follow these steps:
1. Understand the Total Setup:
- Total number of cards: [tex]\(52\)[/tex]
- Total number of aces: [tex]\(4\)[/tex]
- Hand size: [tex]\(5\)[/tex]
2. Calculate the Number of Favorable Outcomes:
- We need exactly 4 aces and 1 other card.
- Number of ways to choose 4 aces from the 4 aces available:
[tex]\[
\binom{4}{4} = 1
\][/tex]
- Number of ways to choose 1 other card from the remaining 48 cards (since we already used the 4 aces):
[tex]\[
\binom{48}{1} = 48
\][/tex]
- Therefore, the number of favorable outcomes (ways to get 4 aces and 1 other card) is:
[tex]\[
1 \times 48 = 48
\][/tex]
3. Calculate the Total Number of Possible Outcomes:
- We need the number of ways to choose any 5 cards out of the total 52 cards:
[tex]\[
\binom{52}{5}
\][/tex]
4. Calculate the Actual Numbers:
- The total number of ways to choose 5 cards from 52 is:
[tex]\[
\binom{52}{5} = 2598960
\][/tex]
5. Determine the Probability:
- The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[
\text{Probability} = \frac{48}{2598960}
\][/tex]
6. Simplify the Probability:
- Performing this division gives us:
[tex]\[
\frac{48}{2598960} \approx 1.846892603195124 \times 10^{-5}
\][/tex]
Thus, the probability that a five-card poker hand dealt to you contains exactly 4 Aces is approximately [tex]\(1.8469 \times 10^{-5}\)[/tex] or in decimal form, approximately [tex]\(0.0000184689\)[/tex].
