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.
Natural Numbers Starts From Which Digit?
Learn where natural numbers start from. Understand the difference between natural numbers (starting from 1) and whole numbers (starting from 0).
Is Every Natural Number a Whole Number?
Find out if every natural number is a whole number (True/False). Understand the difference between natural numbers (N) and whole numbers (W) in Class 9 Math.
Neon Number — Definition, Examples and How to Check
A neon number is a number whose sum of the digits of its square equals the number itself. Learn the definition, examples (like 9) and how to check a neon number.
Common Number and Unit Conversions (Indian System)
Learn all important number conversions — lakh, crore, million, billion. Also understand unit conversions for length, area, time, and weight.
Formula for Number of Diagonals in a Polygon
Learn the formula for the number of diagonals in any polygon. Solve for triangle, quadrilateral, pentagon, hexagon with step-by-step examples.
Turn this guide into revision flashcards, a practice exam, or an AI-generated podcast — free, no signup required.