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Complete IGCSE Edexcel guide to calculating probability, drawing tree diagrams, using sample space diagrams, and finding expected frequency. Worked examples for grades 5-7.
Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event) = number of favourable outcomes divided by total number of outcomes. For combined events, use tree diagrams (multiply along branches, add between branches) or sample space diagrams. Expected frequency = probability multiplied by the number of trials.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
Probability ranges from 0 (impossible) to 1 (certain). For equally likely outcomes, the probability of an event is the number of favourable outcomes divided by the total number of possible outcomes.
P(event) = favourable outcomes / total outcomes
P(not event) = 1 - P(event)
Probabilities can be expressed as fractions, decimals, or percentages. In IGCSE, fractions are preferred unless the question specifies otherwise. Always simplify fractions where possible.
A sample space diagram is a grid that shows all possible outcomes of two combined events. One event is listed along the top and the other down the side. Each cell shows the combined outcome. This makes it easy to count favourable outcomes.
For example, a sample space for rolling two dice would be a 6 by 6 grid showing all 36 possible outcomes. To find P(total = 7), count the cells that sum to 7 (there are 6) and divide by 36, giving P = 6/36 = 1/6.
A tree diagram displays the outcomes of sequential events. Each event creates a set of branches, with probabilities written on each branch. The two key rules are:
| Rule | Operation | When to Use |
|---|---|---|
| AND rule | Multiply along branches | Finding P(A and B) — one specific path |
| OR rule | Add between branches | Finding P(A or B) — multiple paths |
All branches from a single node must sum to 1. After calculating the probabilities for all possible final outcomes, check that they sum to 1 as a verification step.
Expected frequency is the predicted number of times an outcome will occur over many trials. It is calculated by multiplying the probability by the number of trials.
Expected frequency = probability x number of trials
The expected frequency may not be a whole number — this is fine because it represents a theoretical average rather than an actual count. In practice, the actual frequency will vary around the expected value.
Relative frequency is the experimental probability based on actual trials: the number of times an event occurred divided by the total number of trials. As the number of trials increases, relative frequency approaches the theoretical probability.
IGCSE questions may ask you to compare relative frequency with theoretical probability, or use relative frequency to estimate probability when the outcomes are not equally likely (e.g. a biased spinner).
A bag contains 3 red, 5 blue, and 2 green balls. A ball is picked at random. Find P(blue) and the expected number of blue balls in 200 picks (with replacement).
Step 1: Total balls = 3 + 5 + 2 = 10.
Step 2: P(blue) = 5/10 = 1/2.
Step 3: Expected blue = (1/2) x 200 = 100.
Answer: P(blue) = 1/2, expected blue in 200 picks = 100
A coin is flipped twice. Draw a tree diagram and find the probability of getting at least one head.
Step 1: Tree diagram branches: Flip 1 (H: 1/2, T: 1/2), Flip 2 (H: 1/2, T: 1/2 from each).
Step 2: Outcomes: HH = 1/4, HT = 1/4, TH = 1/4, TT = 1/4. Check: 1/4 + 1/4 + 1/4 + 1/4 = 1.
Step 3: P(at least one H) = P(HH) + P(HT) + P(TH) = 1/4 + 1/4 + 1/4 = 3/4.
Alternative: P(at least one H) = 1 - P(no heads) = 1 - P(TT) = 1 - 1/4 = 3/4.
Answer: P(at least one head) = 3/4
A spinner has P(red) = 0.3, P(blue) = 0.5, P(green) = 0.2. It is spun 80 times. The results are red: 28, blue: 36, green: 16. Compare the experimental results with expected frequencies.
Expected: Red = 0.3 x 80 = 24, Blue = 0.5 x 80 = 40, Green = 0.2 x 80 = 16.
Comparison: Red occurred 28 times (expected 24, more than expected). Blue occurred 36 times (expected 40, less than expected). Green occurred exactly 16 times (matches expected).
Conclusion: The differences between experimental and expected frequencies are small, which is expected with 80 trials. With more trials, the relative frequencies would likely get closer to the theoretical probabilities.
Answer: Small differences are normal; more trials would bring results closer to expected values.
Adding instead of multiplying along branches: For combined events (A and B), multiply the probabilities along the path. Adding gives P(A or B), which is a different question.
Branches not summing to 1: All branches from any single node must add up to 1. If they do not, there is an error in your probabilities.
Giving probability greater than 1: Probability can never exceed 1. If your answer is greater than 1, check your working for errors.
Not using 1 - P for "at least one" questions: P(at least one) = 1 - P(none). This is almost always easier than listing all the favourable outcomes individually.
Check all outcomes sum to 1: After completing a tree diagram, add all final probabilities. If they sum to 1, your tree is correct. If not, find and fix the error.
Use fractions throughout: Fractions are easier to multiply and add without rounding errors. Only convert to decimals at the very end if the question requires it.
Label tree diagrams clearly: Write the event name at the start of each set of branches and the outcome at the end of each branch. This prevents confusion in complex trees.
Use "1 minus" for complement questions: When asked for "at least one", "not all the same", or similar, use P = 1 - P(opposite event). This shortcut saves time and reduces errors.
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