Question

From a deck of $52$ cards, two cards are missing. One card is drawn at random from the remaining cards and found to be a spade. Then which of the following is/are correct ?

• A. Probability that both the missing cards were spades is $\frac{22}{425}$
• B. Probability that one missing card was spade is $\frac{156}{425}$
• C. Probability that neither of the missing cards was spade is $\frac{247}{425}$
• D. Probability that both the missing cards were same suit is $\frac{4}{17}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Probability that both the missing cards were spades is $\frac{22}{425}$
B Probability that one missing card was spade is $\frac{156}{425}$
C Probability that neither of the missing cards was spade is $\frac{247}{425}$
D Probability that both the missing cards were same suit is $\frac{4}{17}$

Probability that both the missing cards were spades is equivalent to drawing three cards from a full deck, such that the first card is a spade and then find the probability that the second and third cards are also spades,
Therefore, required probability is,
$P\left(\text{both spades}\right)=\frac{12}{51}\cdot \frac{11}{50}=\frac{22}{425}$

Similarly,
Probability that one missing card was spade is,
$P\left(\text{one spade}\right)=\frac{12}{51}\cdot \frac{39}{50}+\frac{39}{51}\cdot \frac{12}{50}=\frac{156}{425}$

Probability that neither of the missing cards was spade is,
$P\left(\text{no spade}\right)=\frac{39}{51}\cdot \frac{38}{50}=\frac{247}{425}$

Probability that both the missing cards were same suit is given as,
$P\left(\text{same suit}\right)=P\left(\text{both spades}\right)+P\left(\text{both clubs}\right)+P\left(\text{both diamonds}\right)+P\left(\text{both hearts}\right)$
$=\frac{12}{51}\cdot \frac{11}{50}+3(\frac{13}{51}\cdot \frac{12}{50})$
$=\frac{4}{17}$