Four cards are drawn successively without replacement from a deck of 52 playing cards. The probability that all the four cards are king is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{1}{270725} \)
There are 4 kings in 52 cards. Drawing without replacement, each draw leaves one king and one card fewer. By the multiplication rule:
\( \displaystyle P=\frac{4}{52}\times\frac{3}{51}\times\frac{2}{50}\times\frac{1}{49} \)
\( \displaystyle =\frac{24}{6497400} \)
\( \displaystyle =\frac{1}{270725} \)
Shortcut: Only the set of 4 cards matters, so \( P=\frac{1}{^{52}C_{4}}=\frac{1}{270725} \), since \( ^{52}C_{4}=\frac{52\cdot 51\cdot 50\cdot 49}{24}=270725 \).
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{28560} \) — Incorrect. The probability is \( \frac{1}{270725} \), not \( \frac{1}{28560} \).
Option (2) \( \displaystyle\frac{1}{28561} \) — Incorrect. \( \frac{1}{28561}=\left(\frac{4}{52}\right)^{4}=\left(\frac{1}{13}\right)^{4} \) is the probability with replacement, where each draw is again from a full deck.
Option (4) \( \displaystyle\frac{1}{27720} \) — Incorrect. The probability is \( \frac{1}{270725} \), not \( \frac{1}{27720} \).
NCERT Exemplar:
Probability
