When a problem states 'an integer is chosen at random between 1 and 100,' the total sample space depends on whether 'between' is inclusive or exclusive. Most NCERT/CBSE problems mean exclusive — integers from 2 to 99, giving 98 total integers. When the problem says 'from 1 to 100,' it is inclusive — 100 integers. Always read the problem carefully. Below are solved examples of the most common probability questions of this type.
'Between 1 and 100' (exclusive) → integers 2 to 99 → n(S) = 98.
'From 1 to 100' (inclusive) → integers 1 to 100 → n(S) = 100.
P(event) = Favourable outcomes / Total outcomes.
Primes between 2 and 99: 25 primes. P = 25/98.
Multiples of 7 between 2 and 99: 14 numbers. P = 14/98 = 1/7.
Perfect squares from 1 to 100: 10 numbers (1,4,9,...,100). P = 1/10.
Multiples of 5 from 1 to 100: 20 numbers. P = 1/5.
Always identify whether 1 and 100 are included or excluded.
Interpretation:
Case 1: 'Between 1 and 100' (exclusive) • Integers: 2, 3, 4, ..., 99 • Total sample space n(S) = 98
Case 2: 'From 1 to 100' or '1 to 100 inclusive' • Integers: 1, 2, 3, ..., 100 • Total sample space n(S) = 100
Case 3: 'Between 1 and 100 inclusive' • Same as Case 2, n(S) = 100
Formula for probability: P(event) = n(favourable outcomes) / n(total outcomes)
For most problems below, we use n(S) = 98 (exclusive 'between 1 and 100'). If your textbook states 'from 1 to 100', use n(S) = 100.
Using n(S) = 98 (integers 2 to 99):
Problem 1: Find P(chosen integer is divisible by 7). Favourable: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98 = 14 numbers P = 14/98 = 1/7
Problem 2: Find P(chosen integer is NOT divisible by 7). P = 1 − 1/7 = 6/7
Problem 3: Find P(integer is a perfect square). Perfect squares between 2 and 99: 4, 9, 16, 25, 36, 49, 64, 81 = 8 numbers P = 8/98 = 4/49
Problem 4: Find P(integer is a prime number). Primes between 2 and 99: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 = 25 primes P = 25/98
Problem 5: Find P(integer is divisible by both 2 and 3, i.e., divisible by 6). Divisible by 6 between 2 and 99: 6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96 = 16 numbers P = 16/98 = 8/49
Using n(S) = 100 (integers 1 to 100):
Problem 6: Find P(integer is divisible by 5). Divisible by 5 from 1 to 100: 5,10,15,...,100 → 100/5 = 20 numbers P = 20/100 = 1/5
Problem 7: Find P(integer is divisible by 8). Divisible by 8 from 1 to 100: 8,16,24,...,96 → floor(100/8) = 12 numbers P = 12/100 = 3/25
Problem 8: Find P(integer is a perfect square from 1 to 100). Perfect squares: 1,4,9,16,25,36,49,64,81,100 = 10 numbers P = 10/100 = 1/10
Problem 9: Find P(integer ends in 0 or 5). Ends in 0: 10,20,...,100 = 10 numbers Ends in 5: 5,15,...,95 = 10 numbers Total = 20 (these are all multiples of 5) P = 20/100 = 1/5
Problem 10: Find P(integer is a two-digit number). Two-digit numbers from 1–100: 10,11,...,99 = 90 numbers P = 90/100 = 9/10
If 'between 1 and 100' is exclusive (most common interpretation): integers are 2, 3, ..., 99 → total = 98. If inclusive (1 to 100): total = 100. Always check your specific textbook problem for the interpretation.
Using exclusive 'between 1 and 100' (n=98): Multiples of 7 from 2 to 99 are: 7,14,21,28,35,42,49,56,63,70,77,84,91,98 = 14 numbers. P(divisible by 7) = 14/98 = 1/7.
Using 1 to 100 inclusive (n=100): Perfect squares from 1 to 100 are: 1,4,9,16,25,36,49,64,81,100 = 10 numbers. P = 10/100 = 1/10.
Using exclusive (n=98): Primes from 2 to 99: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 = 25 primes. P = 25/98.
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