If two cards are drawn at random without replacement from a pack of 52 playing cards, then the probability that both the cards are red, is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 25/102
A pack has 26 red cards. Drawing without replacement, the multiplication rule gives
\( \displaystyle P(\text{both red})=\frac{26}{52}\times\frac{25}{51} \)
\( \displaystyle =\frac{1}{2}\times\frac{25}{51} \)
\( \displaystyle =\frac{25}{102} \)
Shortcut: Using combinations: \( \frac{^{26}C_{2}}{^{52}C_{2}}=\frac{325}{1326}=\frac{25}{102} \).
Why the other options are incorrect
Option (1) 1/4 — Incorrect. \( \frac{1}{4}=\frac{26}{52}\times\frac{26}{52} \) is the probability with replacement; the second card is drawn from only 51 cards, of which 25 are red.
Option (2) 1/17 — Incorrect. \( \frac{1}{17}=\frac{13}{52}\times\frac{12}{51} \) is the probability that both cards are of one given suit, such as hearts, not both red.
Option (3) 13/51 — Incorrect. \( \frac{13}{51}=\frac{26}{52}\times\frac{26}{51} \) reduces the total to 51 but still counts 26 red cards for the second draw; only 25 remain.