Two cards are drawn from a deck of cards. The probability that both are of same suit is

Options

A. `1/2`
B. `1/13`
C. `4/17`
D. None of these

Answer

To solve the problem of finding the probability that both cards drawn from a deck are of the same suit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Deck Composition**: A standard deck of cards has 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards.

2. **Determine the Total Ways to Choose 2 Cards**: The total number of ways to choose 2 cards from a deck of 52 cards can be calculated using the combination formula:
\[ \text{Total ways} = \binom{52}{2} = \frac{52!}{2!(52-2)!} = \frac{52 \times 51}{2 \times 1} = 1326 \]

3. **Calculate Ways to Choose 2 Cards from the Same Suit**: For each suit, the number of ways to choose 2 cards from the 13 cards in that suit is:
\[ \text{Ways for one suit} = \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78 \]

4. **Account for All Suits**: Since there are 4 suits, the total number of ways to choose 2 cards from any of the suits is:
\[ \text{Total ways for all suits} = 4 \times 78 = 312 \]

5. **Calculate the Probability**: The probability that both cards drawn are of the same suit is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{same suit}) = \frac{\text{Total ways for all suits}}{\text{Total ways}} = \frac{312}{1326} \]

6. **Simplify the Probability**: To simplify \(\frac{312}{1326}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 78:
\[ P(\text{same suit}) = \frac{312 \div 78}{1326 \div 78} = \frac{4}{17} \]

### Final Answer:
The probability that both cards drawn are of the same suit is \(\frac{4}{17}\).