Guessing a Random Card

Solution & Explanation

When trying to predict the outcome of randomly drawing a card from a standard deck of 52 playing cards, we can calculate the probabilities for different scenarios: guessing the suit, the rank (or ordinal), and the exact card itself.

a. Probability of Guessing the Suit

• Total Suits: There are 4 suits in a standard deck of cards: hearts, diamonds, clubs, and spades.
• Probability: Since each suit is equally likely to be drawn, the probability $P(\text{suit})$ of correctly guessing the suit of a randomly drawn card is given by: $P(\text{suit}) = \frac{1}{4}$

b. Probability of Guessing the Rank (Ordinal)

• Total Ranks: Each suit contains 13 ranks: Ace, 2 through 10, Jack, Queen, and King.
• Probability: Since each rank is equally likely to be drawn, the probability $P(\text{rank})$ of correctly guessing the rank of a randomly drawn card is: $P(\text{rank}) = \frac{1}{13}$

c. Probability of Guessing the Exact Card

• Total Cards: A standard deck contains 52 unique cards.
• Probability: Since each card is equally likely to be drawn, the probability $P(\text{exact card})$ of correctly guessing the exact card (both suit and rank) is: $P(\text{exact card}) = \frac{1}{52}$

Explanation

1. Suit Probability:

   • With 4 suits available, the likelihood of guessing the correct suit is 1 out of 4.
   • This is a straightforward calculation as each suit (hearts, diamonds, clubs, spades) is equally probable.

2. Rank Probability:

   • With 13 possible ranks, the likelihood of guessing the correct rank is 1 out of 13.
   • Each rank, from Ace to King, has an equal chance of being the correct guess.

3. Exact Card Probability:

   • To guess the exact card, you need to be correct about both the suit and the rank.
   • The probability of guessing the suit and rank simultaneously is the product of the individual probabilities: $P(\text{exact card}) = P(\text{suit}) \times P(\text{rank}) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52}$
   • This confirms that each card in the deck is unique, and thus, the probability of drawing any specific card is 1 in 52.

In summary, the probabilities are derived from the basic principle of equally likely outcomes in a finite sample space, and these calculations illustrate the fundamental concepts of probability in a practical scenario.