Four cards are drawn from a full pack of cards . Find the probability that
at least one of the four cards is an ace.

Answer

To find the probability that at least one of the four cards drawn from a full pack of cards is an ace, we can use the complementary probability approach. This means we will first calculate the probability that none of the four cards drawn is an ace and then subtract that from 1.

### Step-by-Step Solution:

1. **Understand the Total Number of Cards**:
A standard deck of cards has 52 cards, which includes 4 aces and 48 non-aces.

2. **Calculate the Total Ways to Draw 4 Cards**:
The total number of ways to choose 4 cards from 52 is given by the combination formula:
\[ \text{Total ways} = \binom{52}{4} \]

3. **Calculate the Ways to Draw 4 Non-Ace Cards**:
The number of ways to choose 4 cards from the 48 non-ace cards is:
\[ \text{Ways to choose 4 non-aces} = \binom{48}{4} \]

4. **Calculate the Probability of Drawing No Aces**:
The probability of drawing 4 cards with none being an ace is the ratio of the ways to choose 4 non-aces to the total ways to choose 4 cards:
\[ P(\text{No Aces}) = \frac{\binom{48}{4}}{\binom{52}{4}} \]

5. **Calculate the Probability of At Least One Ace**:
The probability of getting at least one ace is the complement of the probability of getting no aces:
\[ P(\text{At least one Ace}) = 1 - P(\text{No Aces}) = 1 - \frac{\binom{48}{4}}{\binom{52}{4}} \]

6. **Calculate the Combinations**:
Now, we need to calculate the values of the combinations:
\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]
\[ \binom{48}{4} = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} = 194580 \]

7. **Substitute the Values**:
Now substitute these values into the probability formula:
\[ P(\text{At least one Ace}) = 1 - \frac{194580}{270725} \]

8. **Calculate the Final Probability**:
Performing the subtraction:
\[ P(\text{At least one Ace}) = 1 - 0.7183 \approx 0.2817 \]

### Final Answer:
The probability that at least one of the four cards drawn is an ace is approximately \(0.2817\) or \(28.17\%\).