a Pair from Two Card Draw

Solution & Explanation

To calculate the likelihood of drawing a pair when selecting two cards from a standard 52-card deck, we need to understand the fundamental principles of probability and combinations.

Step-by-Step Explanation:

1. Understanding the Deck:

   • A standard deck contains 52 cards, divided into 4 suits: hearts, diamonds, clubs, and spades.
   • Each suit has 13 ranks: Ace, 2, 3, ..., 10, Jack, Queen, King.
   • There are 4 cards of each rank (one in each suit).

2. Combination of Two Cards:

   • The total number of ways to choose 2 cards from a 52-card deck is given by the combination formula $\binom{52}{2}$:

     $\binom{52}{2} = \frac{52 \times 51}{2} = 1326$

3. Forming a Pair:

   • A pair consists of two cards of the same rank.

   • For any given rank, there are 4 cards available. The number of ways to choose 2 cards of the same rank from these 4 cards is given by $\binom{4}{2}$:

     $\binom{4}{2} = \frac{4 \times 3}{2} = 6$

4. Total Pairs in the Deck:

   • There are 13 different ranks in the deck.

   • Therefore, the total number of ways to draw pairs from the whole deck is:

     $13 \times 6 = 78$

5. Probability of Drawing a Pair:

   • The probability of drawing a pair is the number of successful outcomes divided by the total number of possible outcomes:

     $P(\text{pair}) = \frac{78}{1326} \approx 0.0588$

6. Simplifying the Probability:

   • The fraction $\frac{78}{1326}$ simplifies to $\frac{1}{17}$.

Conclusion:

The probability of drawing a pair when selecting two cards from a standard 52-card deck is $\frac{1}{17}$ or approximately 5.88%. This result is derived by considering the number of ways to form a pair and the total number of possible two-card combinations from the deck.