Distinct Card Draw

Solution & Explanation

To solve the problem of determining the probability that the second card drawn is either of a different color or suit than the first card, we need to consider the principles of probability, particularly the concept of mutually exclusive and non-exclusive events.

Understanding the Problem:

When drawing two cards sequentially from a standard 52-card deck without replacement, the probability that the second card is of a different color or suit than the first card can be approached by considering two main events:

• Event A: The second card is of a different color than the first.
• Event B: The second card is of a different suit than the first.

The task is to find $P(A \cup B)$, the probability that either event A or event B occurs.

Step-by-Step Solution:

1. Probability of Event A (Different Color):

   • When the first card is drawn, it can be either red or black. Let's assume the first card is red. There are 26 black cards remaining out of the total 51 cards left in the deck.
   • Therefore, the probability of drawing a card of a different color is: $P(A) = \frac{26}{51}.$

2. Probability of Event B (Different Suit):

   • A standard deck has four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards.
   • If the first card drawn is from a particular suit, there are 39 cards left in the deck that are of a different suit (13 cards each from the other three suits).
   • Thus, the probability of drawing a card of a different suit is: $P(B) = \frac{39}{51}.$

3. Probability of Both Events (Different Color and Suit):

   • For a card to be of a different color and suit, it must be from one of the two suits of the opposite color.
   • If the first card is red (hearts or diamonds), the second card must be black (clubs or spades), and vice versa.
   • Therefore, the probability of drawing a card of both a different color and suit is: $P(A \cap B) = \frac{26}{51}.$

4. Applying the Inclusion-Exclusion Principle:

   • To find $P(A \cup B)$, we use the formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B).$
   • Substituting the values: $P(A \cup B) = \frac{26}{51} + \frac{39}{51} - \frac{26}{51} = \frac{39}{51}.$

Conclusion:

The probability that the second card drawn is either of a different color or suit than the first card is $\frac{39}{51}$ or approximately 0.7647. This solution demonstrates the application of basic probability principles and the inclusion-exclusion principle to solve a common type of problem encountered in data science interviews.