Two cards are drawn simultaneously from a well shuffled pack of 52 cards. If X denotes the number of Spades in two draws, then
Match the List-I with List-II
| List-I | List-II | ||
| (A) | P(X = 0) | (I) | 1 |
| (B) | P(X = 1) | (II) | \( \displaystyle\frac{19}{34} \) |
| (C) | P(X = 2) | (III) | \( \displaystyle\frac{13}{34} \) |
| (D) | P(X < 3) | (IV) | \( \displaystyle\frac{1}{17} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (II), (B) - (III), (C) - (IV), (D) - (I)
A pack has 13 spades and 39 other cards. The number of ways of drawing 2 cards is
\( \displaystyle ^{52}C_{2}=\frac{52\times 51}{2}=1326 \)
(A) P(X = 0)
This matches (II) \( \frac{19}{34} \).
\( \displaystyle P(X=0)=\frac{^{39}C_{2}}{^{52}C_{2}} \)
\( \displaystyle =\frac{741}{1326} \)
\( \displaystyle =\frac{19}{34} \)
(B) P(X = 1)
This matches (III) \( \frac{13}{34} \).
\( \displaystyle P(X=1)=\frac{^{13}C_{1}\times{}^{39}C_{1}}{^{52}C_{2}} \)
\( \displaystyle =\frac{507}{1326} \)
\( \displaystyle =\frac{13}{34} \)
(C) P(X = 2)
This matches (IV) \( \frac{1}{17} \).
\( \displaystyle P(X=2)=\frac{^{13}C_{2}}{^{52}C_{2}} \)
\( \displaystyle =\frac{78}{1326} \)
\( \displaystyle =\frac{1}{17} \)
(D) P(X < 3)
This matches (I) 1.
With only two cards drawn, X can be 0, 1 or 2, so X < 3 is a sure event.
\( \displaystyle P(X\lt 3)=\frac{19}{34}+\frac{13}{34}+\frac{2}{34} \)
\( \displaystyle =1 \)
So the matching is (A) - (II), (B) - (III), (C) - (IV), (D) - (I).
Shortcut: X < 3 is a sure event, so (D) matches (I) 1 at once. Only Option (2) has (D) - (I).