Probability tree diagrams show all possible outcomes of multi-step chance events in an organized way. This cheat sheet helps students build trees, label branches, and calculate probabilities without missing outcomes. It is useful for problems involving coins, dice, spinners, cards, surveys, and dependent or independent events.
Students in grades 6-10 can use it as a quick reference for homework, review, or test preparation.
The most important idea is that each path through the tree represents one complete outcome. To find the probability of a single path, multiply the probabilities along its branches, such as . To find the probability of several acceptable paths, add the probabilities of those paths.
For dependent events, branch probabilities can change after each outcome, so labels must be updated carefully.
Key Facts
- A probability tree starts with one point and branches out for each possible outcome at each stage.
- The probabilities on all branches coming from the same point must add to .
- For one complete path, multiply along the branches using when the events are independent.
- For dependent events, multiply using .
- To find the probability of one of several possible outcomes, add the matching path probabilities using when the outcomes do not overlap.
- For complementary events, use .
- For equally likely outcomes, probability can be written as .
- A tree diagram is complete when every possible path from start to finish represents exactly one outcome.
Vocabulary
- Probability tree diagram
- A diagram that uses branches to show all possible outcomes and their probabilities for a multi-step event.
- Branch
- One line in a probability tree that represents a possible outcome at one stage.
- Path
- A complete route through a tree from the start to the final outcome.
- Independent events
- Events where the result of one event does not change the probability of the next event.
- Dependent events
- Events where the result of one event changes the probability of a later event.
- Conditional probability
- The probability that an event happens given that another event has already happened, written as .
Common Mistakes to Avoid
- Adding branch probabilities along one path instead of multiplying them is wrong because a path represents events happening together, so use multiplication.
- Forgetting to add all matching paths is wrong because an event may happen in more than one way in the tree.
- Using the same probabilities after an item is removed is wrong for dependent events because the total number of possible outcomes changes.
- Letting branches from one point add to more or less than is wrong because one of those outcomes must occur.
- Counting the same path twice is wrong because each complete path should represent one unique final outcome.
Practice Questions
- 1 A coin is flipped twice. Use a probability tree to find .
- 2 A bag has red marbles and blue marbles. One marble is chosen, replaced, and then another is chosen. Find .
- 3 A bag has green marbles and yellow marbles. Two marbles are chosen without replacement. Find .
- 4 Explain why the branch probabilities change in a tree diagram when two cards are drawn from a deck without replacement.
Understanding Probability Tree Diagrams
A tree is more than a drawing. It is a model of a process that happens in stages. The first set of branches describes what can happen first.
Each later set describes what can happen after a particular earlier result. This order matters.
For example, choosing a colored counter first and then choosing another counter creates a different situation from choosing two counters at the same time. A tree keeps the order visible, which makes the full sample space easier to track.
The key decision is whether an earlier result changes the later chances. Replacement is a useful clue. If a counter is returned to a bag before the next draw, the contents of the bag stay the same.
The later branch probabilities remain unchanged. If it is not returned, there is one fewer counter in the bag and the total number available has changed.
The second set of branches must then use new fractions. This is why a tree for drawing cards without replacement has different labels after each first card result.
Trees can work with counts before they work with probabilities. Suppose a class survey records transport to school and year group. You can place the number of students at each branch, then turn each count into a fraction of the relevant group.
This helps explain conditional probability. If the condition says that a student takes the bus, ignore paths for students using other transport. The remaining paths form a smaller group.
The required chance is the number of suitable students divided by the number in that smaller group. A conditional probability is therefore a change in the group being considered.
In real situations, tree diagrams help people compare routes through a process. A factory may sort an item as passing or failing an inspection, then test it again. A medical screening result may be positive or negative, followed by a more accurate test.
In these cases, a result that seems likely overall can mean something different after earlier information is known. Trees make this clear because every path carries both the chance of reaching that point and the history that led there. They are especially useful when words such as given, after, without replacement, or at least appear in a problem.
Careful checking prevents most mistakes. Make sure every final result appears once, not twice and not missing. Check each split separately, since probabilities leaving that point should describe all possibilities from there.
Do not add branch labels from different stages when finding a complete route. Find the chance of each full route first, then combine only the routes that fit the event.
For an at least condition, it is often quicker to find the opposite outcome and subtract its probability from one. Writing a short label beside each final path can stop you from adding a path that does not actually match the wording.