What's the chance of accurately guessing a card I draw from a deck? If I let you ask whether it's red or a face card to boost your odds, which query would you select?

Question Analysis

The question requires understanding the probability of an event involving a standard deck of cards and how to use strategic queries to increase the chances of guessing accurately. Initially, it asks for the probability of guessing a randomly drawn card without any additional information. Then, it introduces the option of asking one question to improve the odds, specifically whether the card is red or a face card. The question tests both probability calculation skills and strategic decision-making.

Answer

To solve this problem, we need to consider a standard deck of 52 cards, consisting of 26 red cards (hearts and diamonds) and 26 black cards (clubs and spades). Among these, there are 12 face cards (3 face cards each in hearts, diamonds, clubs, and spades).

Initial Probability

• Probability of guessing the card right without additional information: Since there are 52 cards, the initial probability of accurately guessing the card is ( \frac{1}{52} ).

Using a Strategic Query

Now, we have the option to ask whether the card is red or a face card, and we must choose the query that maximizes our chances:

• Red Card Query:
  • There are 26 red cards. If the card is red, our pool is reduced to 26 cards.
  • Probability of guessing correctly if the card is red: ( \frac{1}{26} ).
• Face Card Query:
  • There are 12 face cards. If the card is a face card, our pool is reduced to 12 cards.
  • Probability of guessing correctly if the card is a face card: ( \frac{1}{12} ).

Optimal Query Choice

Given the options, asking if the card is a "face card" significantly reduces the pool to 12 cards, thereby increasing our probability of guessing correctly to ( \frac{1}{12} ), which is higher than ( \frac{1}{26} ).

Conclusion: To maximize the probability of accurately guessing the card, choose to ask if the card is a face card.