Question:
If three cards are drawn randomly from a well-shuffled pack of \(52\) cards, then the probability that all the three bear a prime number is
If three cards are drawn randomly from a well-shuffled pack of \(52\) cards, then the probability that all the three bear a prime number is
Show Hint
When cards are drawn without replacement, use combinations:
\[
P=\frac{\binom{\text{favourable cards}}{r}}
{\binom{\text{total cards}}{r}}.
\]
First count the required cards carefully, then apply combinations directly.
Updated On: Jul 9, 2026
- \(\dfrac{35}{1105}\)
- \(\dfrac{28}{1105}\)
- \(\dfrac{21}{1105}\)
- \(\dfrac{18}{1105}\) \bigskip
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The Correct Option is B
Solution and Explanation
Concept:
The probability of an event is
\[
P(E)=\frac{\text{Number of favourable outcomes}}
{\text{Total number of outcomes}}.
\]
Since cards are drawn without replacement, combinations are used.
Step 1: Count the cards bearing prime numbers. In a standard deck, the numbered cards are \[ 2,3,4,5,6,7,8,9,10. \] The prime numbers among them are \[ 2,\;3,\;5,\;7. \] Each occurs in \(4\) suits. Hence the total number of prime-numbered cards is \[ 4\times 4=16. \]
Step 2: Find the number of favourable outcomes. To draw \(3\) cards, all bearing prime numbers, \[ \text{Favourable outcomes} = \binom{16}{3}. \] \[ = \frac{16\cdot15\cdot14}{3\cdot2\cdot1} = 560. \]
Step 3: Find the total number of outcomes. The total number of ways of drawing \(3\) cards from \(52\) cards is \[ \binom{52}{3}. \] \[ = \frac{52\cdot51\cdot50}{3\cdot2\cdot1} = 22100. \]
Step 4: Compute the probability. \[ P = \frac{\binom{16}{3}} {\binom{52}{3}} = \frac{560}{22100}. \] Dividing numerator and denominator by \(20\), \[ P = \frac{28}{1105}. \]
Step 5: Write the final answer. \[ \boxed{\frac{28}{1105}} \]
Step 1: Count the cards bearing prime numbers. In a standard deck, the numbered cards are \[ 2,3,4,5,6,7,8,9,10. \] The prime numbers among them are \[ 2,\;3,\;5,\;7. \] Each occurs in \(4\) suits. Hence the total number of prime-numbered cards is \[ 4\times 4=16. \]
Step 2: Find the number of favourable outcomes. To draw \(3\) cards, all bearing prime numbers, \[ \text{Favourable outcomes} = \binom{16}{3}. \] \[ = \frac{16\cdot15\cdot14}{3\cdot2\cdot1} = 560. \]
Step 3: Find the total number of outcomes. The total number of ways of drawing \(3\) cards from \(52\) cards is \[ \binom{52}{3}. \] \[ = \frac{52\cdot51\cdot50}{3\cdot2\cdot1} = 22100. \]
Step 4: Compute the probability. \[ P = \frac{\binom{16}{3}} {\binom{52}{3}} = \frac{560}{22100}. \] Dividing numerator and denominator by \(20\), \[ P = \frac{28}{1105}. \]
Step 5: Write the final answer. \[ \boxed{\frac{28}{1105}} \]
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