Probability of Combined Events · Form 4
How a tree diagram organises combined events
A tree diagram draws each stage of a combined experiment as a set of branches, so every possible path, and its probability, is laid out and easy to see. You multiply along a path to get that outcome's probability.
Stages become branches
Each stage of the experiment is a point that splits into branches, one branch for each possible outcome of that stage. A path from the start to the end, read left to right, is one combined outcome, for two coins, the top path H then H is the outcome HH.
The probability of each step is written on its branch, so the whole experiment is visible in one picture.
Nothing gets missed, and the rules show up
Because every branch is drawn, a tree makes it hard to leave out an outcome, the main reason it beats listing from memory. The branches leaving any one point always add up to 1, which is a quick check that you haven't dropped a case.
Following a single path and multiplying its branch probabilities gives that path's probability, and this is exactly where the multiplication idea for combined events lives.
When to reach for one, and the classic error
A tree diagram fits when an experiment happens in stages, do this, then that, especially two or three draws or tosses in a row. The classic error is adding along a path instead of multiplying: along one path you multiply, and you only add when you combine the probabilities of several complete paths.
Watch also that for dependent events the second set of branches carries different numbers depending on which first branch you took, so the branches are not simply copied down.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What exactly do the branches and numbers on a tree diagram represent?
Each branch represents one possible outcome at that stage, and the number on it is the probability of that outcome given everything that happened on the branches before it. Branches from the same point must have probabilities that add to 1.
How do I find the probability of a combined outcome from a tree diagram?
Multiply the probabilities along the single path that leads to that outcome. If a question asks for two or more different combined outcomes (e.g., "at least one"), find each qualifying path's probability separately, then add those path probabilities together.
What's a common error when reading probabilities off a tree diagram?
Forgetting to adjust later-branch probabilities when events are dependent (without replacement), or adding along a path instead of multiplying. Also, mixing up "and" (multiply along one path) with "or" (add different paths) loses marks even when the diagram itself is correct.