Probability 2 (Edexcel GCSE Maths): Revision Notes
Probability 2
The fundamental probability rule
When looking at any event with multiple possible outcomes, there's one crucial rule to remember: all probabilities must add up to 1. This means if you list every possible outcome and add their probabilities together, the total will always equal 1.
The Fundamental Probability Rule
For any event with multiple possible outcomes:
This rule is the foundation for solving many probability problems, especially when you need to find missing probability values.
Example: Spinner with Three Colors
If a spinner can land on red, yellow, or green sections:
No matter what the individual probabilities are, they must always sum to exactly 1.
Complementary probability
Complementary probability helps you find the likelihood that something will NOT happen when you already know the probability that it WILL happen.
The formula is:
This approach is particularly useful with dice problems where it's often easier to calculate the probability of something not happening first.
Example: Rolling a Six
If the probability of rolling a 6 on a fair dice is , then the probability of NOT rolling a 6 is:
This makes sense because there are 5 ways to not roll a 6 (rolling 1, 2, 3, 4, or 5) out of 6 possible outcomes.
Finding unknown probabilities
You can use algebra to find missing probability values. When probabilities are given as expressions involving unknowns (like x), set up an equation using the fundamental rule.
Here's how to approach these problems:
- Write out all the probabilities and set their sum equal to 1
- Collect like terms together
- Solve the equation to find the unknown value
- Always check your answer by substituting back
Worked Example: Finding Unknown Probability
If a spinner has sections with probabilities 0.12, , , and 0.28:
Step 1: Set up the equation using the fundamental rule
Step 2: Collect like terms
Step 3: Solve for x
Step 4: Check the answer ✓
Expectation
Expectation tells you how many times you can expect something to happen over multiple trials. It gives you a theoretical prediction, though the actual results may vary slightly.
The formula is:
Expectation gives you a theoretical average - the actual results will likely be close to this value but probably won't match it exactly. The more trials you perform, the closer your actual results should be to the expected value.
Example: Coin Flips
If you flip a coin 100 times, you can expect about 50 heads:
However, you probably won't get exactly 50 heads - you might get 48, 52, or some other number close to 50.
Fair and biassed objects
You can use expectation to determine whether dice, coins, or other objects are fair (unbiased) or biassed.
- A fair object gives results close to what you'd expect
- A biased object gives results that differ significantly from expectation
Example: Testing a Coin for Bias
If you flip a coin 50 times:
- Expected heads for a fair coin =
- If you get about 25 heads (maybe 23-27), the coin is probably fair
- If you get significantly more or fewer heads (like 15 or 35), the coin is probably biased
Sample Size Matters
The larger your sample size, the more confident you can be about whether an object is fair or biassed. A few unusual results don't necessarily indicate bias, but consistent patterns over many trials do.
Key Points to Remember:
- All probabilities for an event must add up to 1
- Use complementary probability:
- Set up equations when solving for unknown probabilities using algebra
- Expectation = Number of trials × Probability gives you theoretical predictions
- Compare actual results with expected results to determine if objects are fair or biassed