Drawing a Pair in Poker

Solution & Explanation

To solve the problem of calculating the probability of obtaining a pair when drawing N cards from a standard 52-card deck, we need to consider the complementary probability of not obtaining a pair, and then subtract this from 1.

Step-by-Step Solution:

1. Understanding the Problem:

• A standard deck has 52 cards, divided into 4 suits and 13 ranks.
• A pair is formed when two cards of the same rank are drawn.
• We need to find the probability of obtaining at least one pair in a hand of N cards.

2. Complementary Probability Approach:

• Instead of directly calculating the probability of obtaining a pair, it's simpler to calculate the probability of not obtaining a pair, and then subtract from 1.
• Let $P(\text{pair}|N)$ be the probability of obtaining at least one pair when drawing N cards.
• $P(\text{pair}|N) = 1 - P(\text{no pair}|N)$

3. Calculating $P(\text{no pair}|N)$:

• To avoid drawing a pair, each card drawn must be of a different rank.

• For the first card, any card can be drawn, so the probability is 1.

• For the second card, it must be of a different rank than the first, so there are 48 valid options out of the remaining 51 cards.

• For the third card, it must be of a different rank than the first two, so there are 44 valid options out of the remaining 50 cards.

• Continuing this pattern, the general formula for $P(\text{no pair}|N)$ when 1 < N <= 13 is:

  $P(\text{no pair}|N) = \frac{52}{52} \times \frac{48}{51} \times \frac{44}{50} \times \frac{40}{49} \times \cdots \times \frac{52 - 4(N-1)}{52 - (N-1)}$

4. Special Cases:

• If N < 2, $P(\text{pair}|N) = 0$ because at least two cards are needed to form a pair.
• If N > 13, $P(\text{pair}|N) = 1$ because with more than 13 cards, at least one pair is guaranteed.

5. Example Calculation:

• For N = 5:

  $P(\text{no pair}|5) = \frac{52}{52} \times \frac{48}{51} \times \frac{44}{50} \times \frac{40}{49} \times \frac{36}{48}$

  • Calculate this product to find the probability of no pair.
  • Subtract from 1 to find $P(\text{pair}|5)$.

Conclusion:

The approach of calculating the complementary probability simplifies the problem and provides a clear path to finding the likelihood of drawing at least one pair in a hand of N cards. The method is based on ensuring each drawn card belongs to a different rank and adjusting for the decreasing number of available cards as each card is drawn.