Key takeaway: A tree diagram is a branching picture that lists every possible outcome of a multi-step activity, with one branch for each choice at each step.
Grade 77.SP.C.8bProbability and Counting
Definition of Tree Diagram
A tree diagram starts at a single point and splits into branches, one for every choice in the first step. From the end of each branch, new branches grow for the second step, and so on. Following a path from the root to the tip spells out one complete outcome.
Counting the tips gives the number of outcomes, and no outcome can be forgotten because every path is drawn. When each branch also carries a probability, multiplying along a path gives the chance of that outcome, and adding the paths that match an event gives the chance of the event. Trees are especially handy when the choices at the second step depend on the first.
Formula and notation
A tree has no formula, but the count of leaves follows a product. If step one has \(m\) choices and step two has \(n\) choices:
\[\text{number of outcomes} = m \times n\]
Write each path as a sequence, such as (turkey, juice).
Key facts
- Each complete path through a tree diagram, from root to tip, is exactly one outcome.
- The number of tips equals the product of the numbers of choices at each stage.
- Branch probabilities are multiplied along a path to get that outcome’s chance.
- The chances at the end of all paths always add up to 1.
Tree Diagram example
A school cafeteria offers a turkey or veggie sandwich and a drink of milk, juice or water. Use a tree diagram to count the outcomes. If Pilar picks both items at random, what is the probability she gets a veggie sandwich with juice?
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Two branches for the sandwich, each followed by three branches for the drink: \(2 \times 3 = 6\) outcomes.
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List them: (turkey, milk), (turkey, juice), (turkey, water), (veggie, milk), (veggie, juice), (veggie, water).
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One path is (veggie, juice): \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).
There are 6 outcomes, and the probability of veggie with juice is \(\frac{1}{6}\).
Common mistake
Watch out
Pilar says the lunch has \(2 + 3 = 5\) possible combinations, because there are 2 sandwiches and 3 drinks.
Why it is wrong
Each sandwich can be paired with every drink, so the choices multiply. Drawing the branches shows 3 drinks hanging off each of the 2 sandwiches, which is 6 paths. Adding would only count the items on the menu, not the combinations.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Probability and Counting