This module covers the reasoning behind probability calculations: listing outcomes that cannot happen together, building a two-stage tree, knowing when probabilities can be multiplied, updating fractions when items are not replaced, and checking that a model adds to 1. These skills turn a wordy question into a short, checkable calculation.
Check the current Cambridge IGCSE Mathematics 0580 syllabus for the exact wording of each content point in your exam year. The habits here stay the same across versions. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to write a probability as a fraction, decimal or percentage, and know that it lies between 0 and 1. You should be comfortable adding and multiplying fractions, including finding a common denominator. The number sense module covers the fraction work if it needs a refresh.
An orienting example
A bag holds 4 red and 6 blue counters. Two counters are taken at random, one after the other, without replacement. Find the probability that both are red.
Step 1, first draw. P(red) = 4/10.
Step 2, update for no replacement. One red has gone, so 3 red remain out of 9. P(red) = 3/9.
Step 3, multiply along the path. 4/10 × 3/9 = 12/90 = 2/15.
Check the whole model. Red then blue = 4/10 × 6/9 = 24/90. Blue then red = 6/10 × 4/9 = 24/90. Blue then blue = 6/10 × 5/9 = 30/90. Adding every path: 12 + 24 + 24 + 30 = 90, so the total is 90/90 = 1.
That one question used a tree, independence (it fails here), a no-replacement update and a total check. It shows how the five lessons fit together.
In which order should you study it?
- Represent mutually exclusive outcomes: the listing habit that decides when you can add.
- Use a two-stage probability tree: multiply along paths, add between paths.
- Explain when multiplication needs independence: knowing when the product rule is allowed.
- Update probabilities after an item is not replaced: the most common source of lost marks in tree questions.
- Check that an outcome model totals one: a final test that catches errors from every earlier step.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Adding overlapping events, such as “even” and “multiple of 3” on the same set of numbers.
- Adding along a branch instead of multiplying, or multiplying between alternative paths.
- Assuming independence without checking, especially with weather or selections from a small group.
- Copying the first-stage fractions into the second stage when nothing is replaced.
- Skipping the total check, so a missing branch goes unnoticed.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before opening the answer, and draw the tree even when you think you can do it in your head. The probability tree and counting board lets you build a tree and compare it with your own. The mistake log and retest queue is useful for recording which error type you made, so you can retest it later with a fresh question.
When you get something wrong, read the routing notes at the end of the practice set and return to the lesson it names. Fix the lesson first, then try a similar question a few days later.
If you want teaching beyond self-study, online one-to-one Mathematics tuition means an experienced teacher reads your written working and finds the habit behind each error.