Probability — Study Notes

Overview

Probability measures how likely an event is to happen. In SOF IMO, probability questions test your ability to count outcomes systematically and compute basic chances. This topic appears in both the Mathematical Reasoning section and occasionally in word problems in Everyday Mathematics.

You must understand two approaches: **experimental probability** (based on actual trials or data) and **theoretical probability** (calculated from equally likely outcomes). Most IMO questions are theoretical, involving dice, coins, cards, spinners, or simple combinatorial scenarios. Master the basic formula, learn to list outcomes carefully, and avoid double-counting or missing cases.

The typical question will ask for P(event) in simplest fraction form. Expect 2–3 probability questions in the exam, ranging from straightforward single-event problems to slightly trickier compound scenarios like "picking two items without replacement" or "at least one" conditions.

Key Concepts

• **Random experiment**: Any action whose outcome cannot be predicted with certainty (e.g., tossing a coin, rolling a die).
• **Sample space (S)**: The set of all possible outcomes. For a die, S = {1, 2, 3, 4, 5, 6}; for two coins, S = {HH, HT, TH, TT}.
• **Event (E)**: A subset of the sample space. Example: getting an even number on a die is E = {2, 4, 6}.
• **Theoretical probability**: P(E) = (Number of favourable outcomes) / (Total number of equally likely outcomes). All outcomes must be equally likely for this formula to work.
• **Experimental probability**: P(E) = (Number of times event occurred) / (Total number of trials). Based on observed data or repeated experiments.
• **Certain event**: Probability = 1 (it always happens).
• **Impossible event**: Probability = 0 (it never happens).
• **Range of probability**: For any event E, 0 ≤ P(E) ≤ 1. Probabilities can be expressed as fractions, decimals, or percentages.

Formulas / Key Facts

1. **Basic probability formula**: P(E) = n(E) / n(S), where n(E) is number of outcomes in E and n(S) is total outcomes. 2. **Complement rule**: P(not E) = 1 − P(E). Useful for "at least one" problems. 3. **Sum of all probabilities**: If events cover the entire sample space and don't overlap, their probabilities sum to 1. 4. **Equally likely outcomes**: Each outcome must have the same chance. A biased coin or weighted die violates this assumption. 5. **Probability of an impossible event**: P(∅) = 0. 6. **Probability of a certain event**: P(S) = 1. 7. **Simplification**: Always reduce your probability fraction to lowest terms. P(E) = 4/8 = 1/2. 8. **Sample space size**: For two independent actions, multiply: two dice give 6 × 6 = 36 outcomes; coin and die give 2 × 6 = 12.

Worked Examples

**Example 1: Single die roll** *A fair die is rolled once. Find the probability of getting a prime number.*