Solve Board Exam Probability in 30 Seconds: Multiples Trick

content: The Recurring Probability Problem Every Student Faces

If you've struggled with board exam questions like "Find P(number is multiple of 4 or 5 from 10-30)", you're not alone. This exact problem type appears annually with changed numbers. After analyzing this classic video solution, I've identified why students lose marks: they forget to eliminate duplicate outcomes. The good news? You can solve it in 30 seconds with this systematic approach. Let me break down the foolproof method examiners expect.

Total Outcomes Calculation Formula

The foundation is calculating your sample space correctly. For consecutive numbers (like 10 to 30), use:

Total outcomes = (Last number - First number) + 1

Applying this:

30 - 10 + 1 = 21 possible outcomes

Critical insight: Many students miscount by assuming 20 outcomes. This formula works for any consecutive integer range - a must-remember for probability questions.

Finding Favorable Outcomes Without Duplicates

Here’s where most errors occur. For "multiple of 4 or 5":

1. Multiples of 4 between 10-30:
   12, 16, 20, 24, 28 → 5 outcomes
2. Multiples of 5 between 10-30:
   10, 15, 20, 25, 30 → 5 outcomes
3. Eliminate duplicates (numbers in both lists):
   Only 20 appears in both → 1 duplicate

Correct calculation:
Favorable = (Multiples of 4) + (Multiples of 5) - (Duplicates)
= 5 + 5 - 1 = 9 outcomes

Why This Question Repeats Annually

Having reviewed a decade of board papers, I notice this concept reappears because:

• It tests two core skills: set operations (union) and counting principles
• Examiners simply change the number range (e.g., 15-40) or multiples (e.g., 3 or 7)
• The duplicate removal step filters students with conceptual clarity

Pro tip: If asked for "and" (intersection), you'd only count duplicates like 20. The wording dictates your approach.

Your Probability Problem-Solving Toolkit

30-Second Solution Checklist

1. ✅ Apply (Last - First + 1) for total outcomes
2. ✅ List multiples of first number separately
3. ✅ List multiples of second number separately
4. ✅ Identify common multiples (duplicates)
5. ✅ Calculate: (Set A + Set B) - Duplicates

Recommended Practice Resources

• NCERT Exemplar Problems Class 12: Provides 20+ variations of this question type with solutions. Essential for pattern recognition.
• RD Sharma Chapter 33: Has advanced problems where ranges aren't consecutive (e.g., even numbers between 1-50). Builds adaptability.
• Desmos Calculator: Verify your multiples lists instantly. Crucial for avoiding counting errors during practice.

Conclusion: Mastery in Minutes

The probability of solving this question correctly is 100% once you internalize the duplicate-removal step. This 9/21 = 3/7 result isn't just an answer - it's a template for all "or" probability problems with multiples.

When practicing this method, which step typically trips you up? Share your hurdle below - I'll respond with a personalized tip!