Drawing a Unique Card

Solution & Explanation

To solve this problem, we need to determine the probability of drawing a card from a deck of 52 cards that is either a different color or a different shape from the card initially drawn. This involves understanding the concepts of probability, specifically the union of two events.

Definitions

• Color: In a standard deck, there are two colors:

  • Red: hearts and diamonds (26 cards)
  • Black: spades and clubs (26 cards)

• Shape (Suit): There are four suits:

  • Hearts (13 cards)
  • Diamonds (13 cards)
  • Clubs (13 cards)
  • Spades (13 cards)

Probability Events

1. Event A: Drawing a card of a different color.
2. Event B: Drawing a card of a different shape.

Calculating Probabilities

• Probability of Event A, P(A):

  • If the first card drawn is red, then the remaining 26 cards are black.
  • Probability of drawing a different color from the remaining deck: $P(A) = \frac{26}{51}$

• Probability of Event B, P(B):

  • If the first card drawn is a heart, then the remaining cards are diamonds, clubs, and spades.
  • Probability of drawing a different shape: $P(B) = \frac{39}{51}$

• Probability of Both Events, P(A and B):

  • This represents drawing a card that is both a different color and a different shape. Since the overlap of different color and different shape is the same as event A (all black cards), this is: $P(A \cap B) = \frac{26}{51}$

Union Probability Formula

The formula for calculating the probability of either event A or event B occurring is given by the union probability formula:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Substituting the values calculated:

$P(A \cup B) = \frac{26}{51} + \frac{39}{51} - \frac{26}{51}$

This simplifies to:

$P(A \cup B) = \frac{39}{51}$

Conclusion

The probability of drawing a card from a shuffled deck that is either a different color or a different shape from the initially drawn card is:

$\frac{39}{51} = \frac{13}{17}$

This means there is a 76.47% chance of drawing a card that is either a different color or a different shape.