Probability 2 (AQA GCSE Maths): Revision Notes
Probability 2
The fundamental rule of probability
Probability measures how likely something is to happen, and there's one crucial rule you must remember: the probabilities of all possible outcomes always sum to 1.
This makes perfect sense when you think about it. Something must happen when you carry out an experiment, so if you add up the chances of every possible outcome, you get certainty (which equals 1).
The Fundamental Rule of Probability
The probabilities (P) of all the different outcomes of an event add up to 1.
This is the most important rule in probability and forms the foundation for solving most probability problems.
For example, when you roll a standard dice, there are six possible outcomes (1, 2, 3, 4, 5, or 6). Each outcome has a probability of , and if you add them all up: .
Complement probability
Sometimes it's easier to work out the probability that something won't happen, rather than the probability that it will. This is called the complement.
The Complement Rule
P(event doesn't happen) = 1 - P(event does happen)
This rule is particularly useful when it's easier to calculate what won't happen than what will happen.
Using our dice example, if you want to find the probability of NOT rolling a 6:
- The probability of rolling a 6 =
- The probability of NOT rolling a 6 =
This is much quicker than adding up P(rolling 1) + P(rolling 2) + P(rolling 3) + P(rolling 4) + P(rolling 5).
Solving for unknown probabilities
When you're given a probability table with missing values, you can use the fundamental rule to find them. Since all probabilities must sum to 1, you can set up an equation and solve for the unknown.
Worked Example: Spinner Probabilities
A spinner can land on red, blue, white, or green. The table shows some probabilities, with blue = x and white = 2x.
Step 1: Set up the equation using the fundamental rule Since all probabilities sum to 1: Red + Blue + White + Green = 1
Step 2: Substitute the known values
Step 3: Simplify and solve
Step 4: Find the final answers Therefore: Blue = 0.2 and White = 0.4
Step 5: Check your answer ✓
Expectation
Expectation tells you roughly how many times you'd expect something to happen if you repeated an experiment many times.
Formula: Expected number of outcomes = number of trials × probability
Understanding Expectation
Expectation gives you a prediction, not a guarantee. If you flip a coin 100 times, you'd expect to get heads about 50 times (100 × 0.5 = 50). You probably won't get exactly 50 heads, but it's a good prediction.
Fair vs biassed experiments
You can use expectation to help decide if a dice or coin is fair (all outcomes equally likely) or biassed (some outcomes more likely than others).
Example: Two coins are flipped 50 times each:
- Coin 1: Gets about 25 heads and 25 tails - probably fair
- Coin 2: Gets 15 heads and 35 tails - probably biassed towards tails
If results are very different from what you'd expect, the object is likely biassed.
Practice problems
You'll often need to complete probability tables by finding missing values. This skill is essential for many probability questions you'll encounter.
Practice Example: Sweet Probabilities
Preti has sweets that are red, yellow, green, or orange. Complete the probability table.
Given: Red = 0.13, Yellow = 0.36, Orange = 0.28 Find: Green = ?
Solution: Using the rule that probabilities sum to 1:
Exam tips
These strategies will help you tackle probability questions successfully in your exams and build confidence with these concepts.
Exam Success Tips
- Always check your probabilities add up to 1
- Use complement probability for "not" questions
- Show your working clearly when solving for unknowns
- Remember that expectation gives an approximate answer, not exact
Key Points to Remember:
- All probabilities sum to 1 - this is the fundamental rule you'll use in every probability question
- Complement probability: P(not happening) = 1 - P(happening) - use this for "not" questions
- Set up equations when finding unknown probabilities using the fact they must sum to 1
- Expectation = trials × probability - use this to predict outcomes over multiple attempts
- Check your answers by ensuring all probabilities in a table add up to 1