Introduction to Probability
Probability quantifies the likelihood of an event occurring, expressed as a fraction, decimal, or percentage between 0 and 1. A probability of 0 means an event is impossible, whilst a probability of 1 indicates certainty. For equally likely outcomes, probability equals the number of favourable outcomes divided by the total number of possible outcomes.
Why it matters
Probability appears throughout daily life and advanced mathematics. Weather forecasters use probability when predicting a 30% chance of rain. Insurance companies calculate premiums based on the probability of accidents or claims. Medical researchers express treatment success rates as probabilities, such as an 85% recovery rate. In gambling, casinos rely on probability to ensure long-term profits — a roulette wheel has 37 slots, giving each number a 137 probability. Students encounter probability in GCSE Mathematics, where it connects to statistics, data analysis, and tree diagrams. Advanced topics like conditional probability and normal distributions build directly on these foundational concepts, making probability essential for A-level Further Maths and university statistics courses.
How to solve introduction to probability
Probability — Introduction
- Probability = number of favourable outcomes ÷ total outcomes.
- P is always between 0 (impossible) and 1 (certain).
- List all possible outcomes before counting.
- P(not A) = 1 − P(A).
Example: Fair die: P(3) = 16. P(not 3) = 56.
Worked examples
A hat contains 5 'Yes' slips and 4 'No' slips. You draw one. What is P(Yes)?
Answer: 59
- Count total slips → 5 + 4 = 9 — All the slips together: 5 + 4 = 9. Each slip is equally likely to be drawn.
- Count favourable (Yes) → Favourable = 5 — There are 5 'Yes' slips in the hat.
- Probability = favourable / total → P(Yes) = 59 = 59 — P(Yes) = 5/9. About 56% chance of drawing Yes.
The weather forecast says it will rain on 3 of the next 7 days. If you pick a random day, what is P(rain)?
Answer: 37
- Count total days → Total = 7 — We're looking at a 7-day forecast. Each day is equally likely to be chosen.
- Count rainy days → Favourable = 3 — 3 days are expected to have rain.
- Calculate probability → P(rain) = 37 = 37 — P(rain) = 3/7. About 43% -- some rain expected.
A die is rolled. What is P(number greater than 4)?
Answer: 13
- List the numbers that are number greater than 4 → 5, 6 (2 outcomes) — Go through 1, 2, 3, 4, 5, 6 and check which ones are number greater than 4: 5, 6. That gives us 2 favourable outcomes.
- Total outcomes on a die → 6 — A standard die always has 6 equally likely outcomes.
- Calculate and simplify → P(number greater than 4) = 26 = 13 — 2 favourable out of 6 total = 2/6 = 1/3 after simplifying.
- Is this likely or unlikely? → less than even — 2/6 is about 33%. Less than half -- it's unlikely.
Common mistakes
- Confusing probability with odds leads to errors like stating P(heads) = 1/2 as '1 to 2' instead of recognising it means a 50% chance
- Adding probabilities incorrectly when finding P(A or B) without checking for overlap, such as calculating P(even or multiple of 3) on a die as 3/6 + 2/6 = 5/6 instead of the correct 4/6
- Forgetting that probabilities must sum to 1 results in errors like claiming P(rain) = 0.4 and P(no rain) = 0.7 instead of P(no rain) = 0.6