Tree diagrams organize multi-step probability situations by showing each possible outcome as a path of branches. This cheat sheet helps students decide whether events are independent or dependent and then label branches correctly. It is useful for problems with coins, spinners, cards, marbles, surveys, and choices made in stages.
A clear tree diagram reduces missed outcomes and helps students calculate probabilities accurately.
The most important idea is that each complete path represents a sequence of events. To find the probability of one path, multiply the probabilities on its branches, such as . For dependent events, the second probability changes after the first event, so use .
To find the probability of several acceptable paths, add the probabilities of those paths.
Key Facts
- In a tree diagram, each branch shows one possible outcome and is labeled with its probability, such as .
- The probabilities on all branches leaving the same point must add to .
- For independent events, the outcome of the first event does not change the probability of the second event, so .
- For dependent events, the outcome of the first event changes the probability of the second event, so .
- The probability of a complete path is found by multiplying branch probabilities: .
- For independent events, the multiplication rule becomes .
- To find the probability of either of two non-overlapping paths, add their probabilities: .
- When drawing without replacement, update the total number of items and the number of favorable items after each draw.
Vocabulary
- Tree diagram
- A branching diagram that shows all possible outcomes of a multi-step probability experiment.
- Independent events
- Events are independent when the result of one event does not change the probability of another event.
- Dependent events
- Events are dependent when the result of one event changes the probability of another event.
- Conditional probability
- Conditional probability is the probability of an event happening given that another event has already happened, written as .
- Branch probability
- A branch probability is the probability written on one branch of a tree diagram.
- Path probability
- A path probability is the product of all branch probabilities along one complete path of a tree diagram.
Common Mistakes to Avoid
- Treating dependent events as independent is wrong because probabilities can change after the first outcome, especially when items are not replaced.
- Forgetting to update the denominator after drawing without replacement is wrong because the total number of items decreases after each draw.
- Adding branch probabilities along one path is wrong because a sequence of events uses multiplication, such as .
- Multiplying probabilities from different paths is wrong because separate acceptable outcomes are combined by addition, not multiplication.
- Leaving branch probabilities unlabeled is a mistake because the diagram cannot be checked or used reliably without every branch probability.
Practice Questions
- 1 A bag has red marbles and blue marbles. One marble is drawn, replaced, and then a second marble is drawn. Use a tree diagram to find .
- 2 A box has green cards and yellow cards. Two cards are drawn without replacement. Use a tree diagram to find .
- 3 A coin is flipped and a spinner with equal sections labeled , , , and is spun. Use a tree diagram to find .
- 4 A student draws two names from a hat without replacement. Explain why the second draw is dependent on the first draw and how the tree diagram should show that change.