Probability
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 394 words at 150 words a minute, plus a minute for each question.
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.
Key ideas
- 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.
Watch out for
- 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.
- Confusing "and" with "or": P(A and B) is usually multiplication (for independent events); P(A or B) requires addition minus overlap. Know the difference.
- 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.
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.
Check yourself: 3 questions
1. A bag contains 5 red balls, 3 blue balls and 2 green balls. If one ball is drawn at random from the bag, what is the probability of drawing a blue ball?
2. A die is thrown once. What is the probability of getting a prime number on the top face?
3. A box contains cards numbered 1 to 50. One card is drawn at random from the box. What is the probability that the number on the card is divisible by both 2 and 5?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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