Basic Probability

Revise Basic Probability for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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Basic Probability

💡 Big idea: Probability measures how likely something is to happen using a number from 0 to 1, and you can work it out either by thinking about all possible outcomes or by running experiments.

Quick Summary

📊 The Probability Scale

0 = impossible, 0.5 = even chance, 1 = certain. Probability is always between 0 and 1.

🎲 How to Calculate

Count the outcomes you want, divide by the total possible outcomes. P(A) = favorable ÷ total

🔄 Complementary Events

If the chance of something happening is p, the chance of it NOT happening is 1 − p.

➕ Mutually Exclusive

If two events can't happen together, add their probabilities: P(A or B) = P(A) + P(B)

🔬 From Experiments

Relative frequency = successful trials ÷ total trials. More trials = better estimate.

🔮 Expected Outcome

Expected frequency = probability × number of trials. How many times you'd expect something.

1. The Foundations of Probability

What Exactly Is Probability?

Probability describes the likelihood of something happening. Instead of saying "probably" or "unlikely," we use a precise number between 0 and 1 to measure it.

Think of it like this: if you're totally certain something will happen, its probability is 1. If you're 100% sure it won't happen, it's 0. Anything in between? That's somewhere on the scale.

💡 Tip: Probabilities can be written as fractions (like ½), decimals (like 0.5), or percentages (like 50%). Exams usually want fractions in simplest form unless told otherwise.

Probability Scale ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 0 0.25 0.5 0.75 1 | | | | | IMPOSSIBLE UNLIKELY EVEN CHANCE LIKELY CERTAIN Examples: - Getting heads on a fair coin: 0.5 - Rolling a 6 on a fair die: 1/6 ≈ 0.167 - The sun rising tomorrow: 1 (certain!) - Rolling a 7 on a standard die: 0 (impossible)

Key Terminology You Need to Know

Experiment: A repeatable activity that produces results. Flipping a coin, rolling a die, drawing a card — all experiments. Trial: One single repetition of an experiment. If you flip a coin 10 times, you have 10 trials. Outcome: A possible result from a single trial. When rolling a die, the outcomes are 1, 2, 3, 4, 5, or 6. Event: One outcome, OR a collection of outcomes. Represented by a capital letter (like A). Example: Event A = rolling an even number (which includes 2, 4, and 6). Sample Space: The set of all possible outcomes for an experiment. For a die: {1, 2, 3, 4, 5, 6}. Written as a list or shown in a table. Fair Event: All outcomes have an equal chance of happening. A fair coin has a 0.5 probability for heads AND tails. A fair die has 1/6 for each number. Biased Event: Outcomes don't have equal chances. A bent coin might land on heads 70% of the time. You can spot bias by comparing theoretical and experimental probabilities.

Remember: n(A)

2. How to Calculate Probabilities (Theoretical)

equally likely

Theoretical Probability Formula

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Also in the full note

3. Complementary Events (Not A)
4. Mutually Exclusive Events (Either/Or)
5. Sample Space Diagrams (Grids)
6. Relative Frequency (From Experiments)
7. Expected Frequency (How Many Times You'd Expect)
📝 What to Memorise
✓ Concepts Checklist
🎯 Exam Tips & Common Mistakes

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Basic Probability