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.*

**Solution:** Sample space S = {1, 2, 3, 4, 5, 6}, so n(S) = 6. Prime numbers on a die: E = {2, 3, 5}, so n(E) = 3. P(prime) = n(E)/n(S) = 3/6 = 1/2.

**Example 2: Deck of cards** *One card is drawn from a standard 52-card deck. What is the probability it is a king or a heart?*

**Solution:** Number of kings = 4, number of hearts = 13. King of hearts is counted in both, so by inclusion-exclusion: favourable outcomes = 4 + 13 − 1 = 16. P(king or heart) = 16/52 = 4/13.

**Example 3: Two coins** *Two fair coins are tossed. Find the probability of getting at least one head.*

**Solution:** Sample space: {HH, HT, TH, TT}, n(S) = 4. Method 1 (direct): At least one head means {HH, HT, TH}, so n(E) = 3. P = 3/4. Method 2 (complement): P(at least one head) = 1 − P(no heads) = 1 − P(TT) = 1 − 1/4 = 3/4.

**Example 4: Experimental probability** *A coin is tossed 80 times. Heads appeared 45 times. What is the experimental probability of heads?*

**Solution:** Experimental P(H) = (Number of heads) / (Total tosses) = 45/80 = 9/16. Note: This differs from theoretical P(H) = 1/2 because the coin was not tossed infinitely many times.

Common Mistakes

1. **Forgetting to simplify**: Writing 6/12 instead of 1/2 is technically correct but loses marks if the question asks for simplest form. Always reduce fractions. 2. **Confusing "and" with "or"**: P(A and B) is usually multiplication (for independent events); P(A or B) requires addition minus overlap. IMO problems at this level rarely ask for compound probabilities, but know the difference. 3. **Double-counting in "or" problems**: When counting king or heart, the king of hearts is in both sets. Subtract the overlap: 4 + 13 − 1 = 16, not 17. 4. **Miscounting the sample space**: For two dice, sample space size is 36, not 12. For a coin and a die, it's 12, not 8. List outcomes systematically or use multiplication principle. 5. **Using the wrong denominator in experimental probability**: Use total trials, not total possible outcomes. If a die is rolled 50 times and 6 appears 9 times, experimental P(6) = 9/50, not 9/6.

Quick Reference

• **P(E) = favourable outcomes / total outcomes** — memorize this cold.
• **0 ≤ P(E) ≤ 1** — probabilities never go outside this range.
• **P(not E) = 1 − P(E)** — complement rule saves time on "at least one" questions.
• **Two independent actions**: multiply their outcome counts (e.g., coin + die = 2 × 6 = 12).
• **Simplify every answer** — 10/20 must become 1/2.
• **Experimental vs theoretical**: Experimental uses trial data; theoretical uses logic and equal likelihood.