If two cards are drawn randomly from a pack of 52 cards , then find probability that at least one drawn cards is king card.

Options

A. `(33)/(221)`
B. `(19)/(221)`
C. `(45)/(221)`
D. `(3)/(17)`

Answer

To find the probability that at least one of the two cards drawn from a standard deck of 52 cards is a king, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Cards and Kings**:
- A standard deck has 52 cards.
- There are 4 kings in the deck.

2. **Define the Event**:
- We want to find the probability of drawing at least one king when two cards are drawn.

3. **Use the Complement Rule**:
- Instead of directly calculating the probability of getting at least one king, we can calculate the probability of the complementary event (drawing no kings) and subtract it from 1.
- So, we need to find \( P(\text{at least one king}) = 1 - P(\text{no kings}) \).

4. **Calculate the Probability of Drawing No Kings**:
- If we are drawing two cards and want none of them to be kings, we can first calculate the number of ways to draw 2 cards from the 48 non-king cards.
- The total number of ways to choose 2 cards from 52 is given by \( \binom{52}{2} \).
- The number of ways to choose 2 cards from the 48 non-king cards is \( \binom{48}{2} \).

5. **Calculate the Combinations**:
- Calculate \( \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \).
- Calculate \( \binom{48}{2} = \frac{48 \times 47}{2} = 1128 \).

6. **Find the Probability of No Kings**:
- The probability of drawing no kings is given by:
\[ P(\text{no kings}) = \frac{\text{Number of ways to choose 2 non-king cards}}{\text{Total ways to choose 2 cards}} = \frac{1128}{1326} \]

7. **Calculate the Probability of At Least One King**:
- Now, substitute this value into the complement formula:
\[ P(\text{at least one king}) = 1 - P(\text{no kings}) = 1 - \frac{1128}{1326} \]
- Simplifying this gives:
\[ P(\text{at least one king}) = \frac{1326 - 1128}{1326} = \frac{198}{1326} \]

8. **Simplify the Fraction**:
- Simplifying \( \frac{198}{1326} \) gives:
\[ \frac{198 \div 66}{1326 \div 66} = \frac{3}{20.1} \approx \frac{33}{221} \]

### Final Answer:
The probability that at least one of the two drawn cards is a king is \( \frac{33}{221} \).