Probability for Math Olympiad: Complete Preparation Guide

Why This Matters

Probability is one of the most intuitive yet tricky topics in Math Olympiads. What makes it challenging is that our natural instincts about chance are often wrong — and Olympiad papers exploit this beautifully.

For Class 10 students, competition problems test your ability to define sample spaces correctly, calculate theoretical probabilities, work with complementary events, and solve multi-step probability puzzles.

Best Strategy

Common Pitfalls

Mistakes:

• Sample space errors — For two dice, there are 36 outcomes (not 12). (3,4) is different from (4,3).
• Probability range — $0 \leq P(E) \leq 1$. If your answer is outside this range, recheck.
• "At least one" problems — Use complement: $P(\text{at least one}) = 1 - P(\text{none})$.
• Playing cards — 52 cards, 4 suits (13 each), 12 face cards, 4 aces. Know these counts cold.

Practice Questions

Try these!

Question 1

Two dice are thrown simultaneously. Find the probability of getting a sum of 9.

Solution: Favorable outcomes: (3,6), (4,5), (5,4), (6,3) = 4 outcomes.
Total outcomes = 36.
$P = \frac{4}{36} = \frac{1}{9}$.

Question 2

A bag contains 5 red and 8 blue balls. One ball is drawn. Find the probability that it is not red.

Solution: $P(\text{not red}) = 1 - P(\text{red}) = 1 - \frac{5}{13} = \frac{8}{13}$.

Question 3

A card is drawn from a deck of 52 cards. Find the probability of getting a face card or a spade.

Solution: Face cards = 12, Spades = 13, Face spades = 3.
$P = \frac{12 + 13 - 3}{52} = \frac{22}{52} = \frac{11}{26}$.

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