Question:
From a well shuffled pack of \(52\) cards, two cards are drawn at random. Then, the probability of both the cards being kings is:
From a well shuffled pack of \(52\) cards, two cards are drawn at random. Then, the probability of both the cards being kings is:
Show Hint
When cards are drawn without replacement, use either combinations
\[
P=\frac{{}^{m}C_{r}}{{}^{n}C_{r}}
\]
or multiply successive probabilities. Both methods give the same result.
Updated On: Jun 26, 2026
- \(\dfrac{1}{15}\)
- \(\dfrac{25}{57}\)
- \(\dfrac{35}{256}\)
- \(\dfrac{1}{221}\)
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The Correct Option is D
Solution and Explanation
Step 1: Find the total number of ways to draw two cards.
A standard deck contains \(52\) cards.
The total number of ways to select \(2\) cards from \(52\) cards is \[ {}^{52}C_{2} = \frac{52\times 51}{2} = 1326 \]
Step 2: Find the number of favourable outcomes.
There are \(4\) kings in a deck.
To obtain two kings, we must choose \(2\) cards from these \(4\) kings. Hence, the number of favourable outcomes is \[ {}^{4}C_{2} = \frac{4\times 3}{2} = 6 \]
Step 3: Apply the probability formula.
Probability is given by \[ P(E) = \frac{\text{Number of favourable outcomes}} {\text{Total number of outcomes}} \] Therefore, \[ P(\text{both cards are kings}) = \frac{{}^{4}C_{2}}{{}^{52}C_{2}} \] \[ = \frac{6}{1326} \] \[ = \frac{1}{221} \]
Step 4: Verification using successive probabilities.
Alternatively, \[ P(\text{first card is a king}) = \frac{4}{52} \] and \[ P(\text{second card is a king}\mid \text{first is a king}) = \frac{3}{51} \] Thus, \[ P = \frac{4}{52}\times\frac{3}{51} = \frac{12}{2652} = \frac{1}{221} \] which confirms the result.
Step 5: Final conclusion.
Hence, the required probability is \[ \boxed{\frac{1}{221}} \] Therefore, the correct option is \[ \boxed{(4)\ \frac{1}{221}} \]
A standard deck contains \(52\) cards.
The total number of ways to select \(2\) cards from \(52\) cards is \[ {}^{52}C_{2} = \frac{52\times 51}{2} = 1326 \]
Step 2: Find the number of favourable outcomes.
There are \(4\) kings in a deck.
To obtain two kings, we must choose \(2\) cards from these \(4\) kings. Hence, the number of favourable outcomes is \[ {}^{4}C_{2} = \frac{4\times 3}{2} = 6 \]
Step 3: Apply the probability formula.
Probability is given by \[ P(E) = \frac{\text{Number of favourable outcomes}} {\text{Total number of outcomes}} \] Therefore, \[ P(\text{both cards are kings}) = \frac{{}^{4}C_{2}}{{}^{52}C_{2}} \] \[ = \frac{6}{1326} \] \[ = \frac{1}{221} \]
Step 4: Verification using successive probabilities.
Alternatively, \[ P(\text{first card is a king}) = \frac{4}{52} \] and \[ P(\text{second card is a king}\mid \text{first is a king}) = \frac{3}{51} \] Thus, \[ P = \frac{4}{52}\times\frac{3}{51} = \frac{12}{2652} = \frac{1}{221} \] which confirms the result.
Step 5: Final conclusion.
Hence, the required probability is \[ \boxed{\frac{1}{221}} \] Therefore, the correct option is \[ \boxed{(4)\ \frac{1}{221}} \]
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