NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Sample spaces and probability

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A game involves drawing one card at random from a standard deck of 52 playing cards. The deck contains 4 suits: hearts, diamonds, clubs, and spades, with 13 cards in each suit.

What is the exact probability of drawing a card that is both a heart and a face card (Jack, Queen, or King)? Express your answer as a simplified fraction.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. A standard deck has 52 cards, with 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, including 3 face cards: Jack, Queen, and King.
  2. The event of interest is drawing a card that is both a heart and a face card. This means the card must be from the hearts suit and be one of the three face cards in that suit.
  3. There are 3 face cards in the hearts suit: Jack of hearts, Queen of hearts, and King of hearts.
  4. The total number of possible outcomes when drawing one card at random is 52, since the deck has 52 distinct cards.
  5. The number of favourable outcomes (hearts that are face cards) is 3.
  6. Probability is calculated as: favourable outcomes divided by total possible outcomes.
  7. Substitute values: probability = 3 / 52.
  8. Simplify the fraction 3/52. The greatest common divisor of 3 and 52 is 1, so the fraction is already in simplest form.
  9. Therefore, the exact probability is 3/52.
  10. Check: 3 face cards in hearts, 52 total cards. 3/52 cannot be simplified further.
  11. The probability of drawing a heart that is also a face card is exactly 3/52.
  12. Final answer confirmed.

Reference answer: 3/52

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