Question

what is the probability of getting exactly one ace in a five card hand from a deck of 52 cards

Answer 1

To find the probability of getting exactly one ace in a five-card hand from a standard deck of 52 cards, we can use combinatorics.

1. **Determine the total number of possible five-card hands**:
The total number of ways to choose 5 cards from a deck of 52 cards is given by the combination formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Here, \( n = 52 \) and \( k = 5 \):

\[
\text{Total hands} = \binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5! \cdot 47!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960
\]

2. **Determine the number of ways to get exactly one ace**:
- Choose 1 ace from the 4 available aces: \( \binom{4}{1} \)
- Choose 4 non-aces from the remaining 48 cards: \( \binom{48}{4} \)

Now we calculate these combinations:

\[
\binom{4}{1} = 4
\]

\[
\binom{48}{4} = \frac{48!}{4!(48-4)!} = \frac{48!}{4! \cdot 44!} = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} = 194,580
\]

3. **Total ways to choose a hand with exactly one ace**:
Combine the combinations from the previous two steps:

\[
\text{Ways to choose exactly 1 ace} = \binom{4}{1} \times \binom{48}{4} = 4 \times 194,580 = 778,320
\]

4. **Calculate the probability**:
The probability \( P \) of getting exactly one ace in a five-card hand is given by the ratio of the favorable outcomes to the total outcomes:

\[
P(\text{exactly 1 ace}) = \frac{\text{Ways to choose exactly 1 ace}}{\text{Total hands}} = \frac{778,320}{2,598,960}
\]

Simplifying the fraction:

\[
P(\text{exactly 1 ace}) \approx 0.2995
\]

So, the probability of getting exactly one ace in a five-card hand from a standard deck of 52 cards is approximately **0.2995** or **29.95%**.