Question:
From the set of numbers
\[
\{1,2,3,4,5,6,7,8,9,10,11,12\},
\]
two numbers are selected at random. The probability that the two numbers selected differ by a prime number is
From the set of numbers
\[
\{1,2,3,4,5,6,7,8,9,10,11,12\},
\]
two numbers are selected at random. The probability that the two numbers selected differ by a prime number is
Show Hint
For difference-based counting problems, count separately for each possible difference and then add the results. This avoids missing cases and double counting.
Updated On: Jun 10, 2026
- \(\frac{16}{33}\)
- \(\frac{1}{11}\)
- \(\frac{3}{11}\)
- \(\frac{11}{24}\)
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The Correct Option is A
Solution and Explanation
Concept:
When two numbers are chosen from a finite set, probability is calculated using
\[
P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}.
\]
Here we must count the pairs whose difference is a prime number.
Step 1: Calculate total number of pairs There are \(12\) numbers. The number of ways of selecting \(2\) numbers is \[ ^{12}C_2 = \frac{12\times11}{2} = 66. \] Hence, \[ \text{Total outcomes}=66. \]
Step 2: Prime differences possible The maximum difference between two selected numbers is \[ 12-1=11. \] Prime numbers not exceeding \(11\) are \[ 2,3,5,7,11. \]
Step 3: Count pairs with difference \(2\) \[ (1,3),(2,4),\ldots,(10,12) \] Number of pairs \[ =10. \]
Step 4: Count pairs with difference \(3\) \[ (1,4),(2,5),\ldots,(9,12) \] Number of pairs \[ =9. \]
Step 5: Count pairs with difference \(5\) \[ (1,6),(2,7),\ldots,(7,12) \] Number of pairs \[ =7. \]
Step 6: Count pairs with difference \(7\) \[ (1,8),(2,9),\ldots,(5,12) \] Number of pairs \[ =5. \]
Step 7: Count pairs with difference \(11\) \[ (1,12) \] Number of pairs \[ =1. \]
Step 8: Total favourable pairs \[ 10+9+7+5+1 = 32. \] Therefore, \[ P = \frac{32}{66} = \frac{16}{33}. \] Hence, \[ \boxed{\frac{16}{33}}. \]
Step 1: Calculate total number of pairs There are \(12\) numbers. The number of ways of selecting \(2\) numbers is \[ ^{12}C_2 = \frac{12\times11}{2} = 66. \] Hence, \[ \text{Total outcomes}=66. \]
Step 2: Prime differences possible The maximum difference between two selected numbers is \[ 12-1=11. \] Prime numbers not exceeding \(11\) are \[ 2,3,5,7,11. \]
Step 3: Count pairs with difference \(2\) \[ (1,3),(2,4),\ldots,(10,12) \] Number of pairs \[ =10. \]
Step 4: Count pairs with difference \(3\) \[ (1,4),(2,5),\ldots,(9,12) \] Number of pairs \[ =9. \]
Step 5: Count pairs with difference \(5\) \[ (1,6),(2,7),\ldots,(7,12) \] Number of pairs \[ =7. \]
Step 6: Count pairs with difference \(7\) \[ (1,8),(2,9),\ldots,(5,12) \] Number of pairs \[ =5. \]
Step 7: Count pairs with difference \(11\) \[ (1,12) \] Number of pairs \[ =1. \]
Step 8: Total favourable pairs \[ 10+9+7+5+1 = 32. \] Therefore, \[ P = \frac{32}{66} = \frac{16}{33}. \] Hence, \[ \boxed{\frac{16}{33}}. \]
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