Matching Suit in Two Card Draw

Solution & Explanation

To solve the problem of calculating the probability of drawing two cards from a standard deck where both cards belong to the same suit, we need to understand the structure of a deck and the probabilities involved in drawing cards.

Structure of a Deck

• A standard deck contains 52 cards.
• There are 4 suits: spades, hearts, diamonds, and clubs.
• Each suit contains 13 cards.

Probability Calculation

1. Draw the First Card:

   • The first card can be of any suit, so the probability of drawing any card is 1 (since it doesn't matter which card it is).

2. Draw the Second Card:

   • After drawing the first card, there are 51 cards left in the deck.
   • To ensure the second card is of the same suit as the first, there are only 12 cards left of the same suit.
   • Thus, the probability of drawing a second card of the same suit is $\frac{12}{51}$.

3. Total Probability:

   • Since the first draw can be of any suit, and there are 4 suits, we multiply the probability of drawing the second card by 4 to account for each suit: $P(\text{two cards of the same suit}) = 4 \times \left( \frac{13}{52} \right) \times \left( \frac{12}{51} \right) = \frac{12}{51} = \frac{4}{17} \approx 0.2353$

Explanation

• First Card Draw:

  • The first card drawn sets the suit for the second card.
  • Since there are 13 cards per suit, the probability of drawing the first card from any suit is $\frac{13}{52}$.

• Second Card Draw:

  • After the first card is drawn, 12 cards remain of the same suit, and the total number of remaining cards is 51.
  • Probability of drawing a second card from the same suit is $\frac{12}{51}$.

• Overall Probability:

  • The overall probability of drawing two cards from the same suit, considering all suits, is $\frac{4}{17}$.

Conclusion

The probability of drawing two cards from a standard deck that belong to the same suit is approximately 23.53%. This result is derived from considering the independent probabilities of drawing each card and ensuring the second card matches the suit of the first.