Multiply ALONG branches, add BETWEEN paths
Probability of a sequence = multiply along its branches. For an event that can happen several ways, add the probabilities of each complete path.
Without replacement changes the second branch
If the item is not replaced, the total and the relevant count both drop by on the second pick, so the second-set probabilities differ from the first.
"At least one" — use the complement
is usually the quickest route.
Drawn from real examiner reports.
Integers on branches, not fractions
Writing whole-number counts on the branches instead of probabilities, or not to the required accuracy.
Nov 2024 / June 2024 Paper 1H (Q13a): integer values on branches.
Denominator unchanged (no replacement)
Using the same denominators on the second pick (treating it as with replacement) when items are not replaced.
June 2024 Papers 1H/1HR (Q16/Q21): denominators left unchanged scored no marks.
Counting only one order for "one of each"
For "one red and one blue" you must add both paths (red-then-blue AND blue-then-red).
June 2024 Paper 1HR (Q21): only one of the two products counted.
Adding probabilities to more than 1
Adding probabilities along a single path (instead of multiplying), giving a value greater than 1.
Nov 2024 / June 2024 Paper 1H (Q13b).
Drop the count too, not just the total
For "without replacement" both numbers change on the second pick: the total falls by 1 and the count of the colour already taken also falls by 1. Reducing only the denominator is a common error.
Branch pair not adding to 1
The two probabilities on a pair of branches from the same point must add to 1. If they do not, a numerator or denominator is wrong — a quick check to make before you multiply along the paths.
Label the second set of branches
Always label the second-pick branches (and their probabilities) — unlabelled branches lose marks even with correct arithmetic.
Multiply first, add last
Work out the probability of each complete path by multiplying along it, write each product down, then add the products of the paths you want. Keeping the two steps separate avoids mixing them up.
Use the complement for "at least one"
For "at least one", compute 1 − P(none) instead of adding several separate paths. One subtraction is far less error-prone than several path products added together.
Probabilities on each pair of branches add to . Write them as fractions or decimals, never as whole-number counts.
On a tree diagram, when do you multiply and when do you add?
A bag has red and blue counters. A counter is taken, its colour noted, and replaced; then a second is taken. Find the probability that both are red.
