A card is drawn at random from a well-shuffled pack of 52 playing cards. The probability that the card is a king or a heart is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{4}{13}\)
The pack has 4 kings and 13 hearts. The king of hearts is in both groups, so by the addition rule it is subtracted once:
\( \displaystyle P(K \cup H) = P(K) + P(H) - P(K \cap H) \)
\( \displaystyle = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} \)
\( \displaystyle = \frac{16}{52} \)
\( \displaystyle = \frac{4}{13} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{17}{52}\) — Incorrect. This is \(\frac{4}{52} + \frac{13}{52}\), which counts the king of hearts twice.
Option (2) \(\displaystyle\frac{1}{52}\) — Incorrect. This is the probability of the king of hearts, a card that is a king and a heart.
Option (3) \(\displaystyle\frac{1}{4}\) — Incorrect. This is the probability of a heart only (13 cards out of 52).