Question

If two cards are drawn at a time from a well shuffled pack of 52 cards, then the probability of getting a result containing only one king and only one spade is

• A. \(\frac{8}{221}\)
• B. \(\frac{49}{1326}\)
• C. \(\frac{12}{221}\)
• D. \(\frac{1}{26}\)

Hint:

In card probability questions involving overlapping conditions such as king and spade, always separate into cases and remember special overlap cards like King of Spades. Direct counting without case division often leads to mistakes.

Answer 1

The Correct Option is C

Concept: For probability questions involving selection of cards from a deck, the standard approach is based on combinations. Important ideas used here are:
• Total outcomes are counted using combinations.
• Favorable outcomes are counted according to conditions in the problem.
• Probability formula: \[ P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
• A standard deck contains:
• 52 total cards
• 4 Kings
• 13 Spades
• 1 King of Spades The condition “only one king and only one spade” means among the two selected cards there should be exactly one king and exactly one spade.

Step 1: Find the total number of ways of drawing two cards.
Two cards are selected simultaneously from a deck of 52 cards. Hence total possible outcomes are \[ ^{52}C_2 \] Calculating this value, \[ ^{52}C_2=\frac{52\times51}{2} \] \[ ^{52}C_2=1326 \] Thus total outcomes are \[ 1326 \]

Step 2: Understand carefully what favorable condition means.
The question says:
• Exactly one king
• Exactly one spade Now we divide into cases. We know among kings there are four kings. Among spades there are thirteen spades. One card belongs to both groups: \[ \text{King of Spades} \] So careful counting is required.

Step 3: Case 1 : Choose a king which is not spade.
There are total 4 kings. One of them is King of Spades. So kings which are not spade: \[ 4-1=3 \] Thus number of ways choosing non-spade king: \[ 3 \] Now choose one spade card which is not king. There are 13 spades total. Removing King of Spades leaves \[ 13-1=12 \] Thus number of ways: \[ 12 \] Total ways in this case become \[ 3\times12=36 \]

Step 4: Case 2 : Choose King of Spades.
Suppose one selected card is King of Spades. This card already contributes
• one king
• one spade To satisfy exactly one king and exactly one spade, second card should be neither a king nor a spade. Cards which are kings or spades: \[ 4+13-1=16 \] Subtracting overlap because King of Spades counted twice. So cards which are neither king nor spade: \[ 52-16=36 \] Hence ways in this case are \[ 36 \]

Step 5: Calculate total favorable outcomes.
Adding both possible cases \[ 36+36=72 \] Thus favorable outcomes are \[ 72 \]

Step 6: Apply probability formula.
Probability is \[ P(E)=\frac{72}{1326} \] Dividing numerator and denominator by 6 \[ P(E)=\frac{12}{221} \] Thus final probability becomes \[ \boxed{\frac{12}{221}} \] This matches option (3).