A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Two cards are drawn at random. The probability that at least one of them is an ace is

Options

A. `9/20`
B. `3/(16)`
C. `1/6`
D. `1/9`

Answer

To find the probability that at least one of the two cards drawn from a pack of cards is an ace, we can use the complementary probability approach. This means we will first calculate the probability that neither of the cards drawn is an ace and then subtract that from 1.

### Step-by-Step Solution:

1. **Identify the Total Number of Cards**:
- There are 4 aces, 4 kings, 4 queens, and 4 jacks.
- Total cards = 4 (Aces) + 4 (Kings) + 4 (Queens) + 4 (Jacks) = 16 cards.

2. **Calculate the Total Ways to Draw 2 Cards**:
- The total number of ways to choose 2 cards from 16 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total ways to choose 2 cards} = \binom{16}{2} = \frac{16 \times 15}{2 \times 1} = 120 \]

3. **Calculate the Number of Ways to Draw 2 Non-Ace Cards**:
- If we want to find the probability that neither card is an ace, we first find the number of non-ace cards.
- There are 12 non-ace cards (4 kings + 4 queens + 4 jacks).
- The number of ways to choose 2 cards from these 12 non-ace cards is:
\[ \text{Ways to choose 2 non-ace cards} = \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66 \]

4. **Calculate the Probability of Drawing 2 Non-Ace Cards**:
- The probability that both cards drawn are non-aces is:
\[ P(\text{both non-aces}) = \frac{\text{Ways to choose 2 non-aces}}{\text{Total ways to choose 2 cards}} = \frac{66}{120} = \frac{11}{20} \]

5. **Calculate the Probability of At Least One Ace**:
- The probability of drawing at least one ace is the complement of drawing no aces:
\[ P(\text{at least one ace}) = 1 - P(\text{both non-aces}) = 1 - \frac{11}{20} = \frac{9}{20} \]

### Final Answer:
The probability that at least one of the two cards drawn is an ace is \( \frac{9}{20} \).