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.