Tree Diagram
In short: Tree diagram refers to a sideways branching sketch of a multi-stage experiment, where each path from the root to a tip is one outcome and its branch chances multiply.
7th GradeProbability7.SP.C.8b
What is Tree Diagram?
A tree diagram grows from left to right. Each stage of an experiment sprouts a set of branches, one per possible result, so every route from the start to a tip spells out one complete outcome.
With chances written on the branches, you multiply along a route to get that outcome’s probability, then add the routes that belong to the event you want. The tips of a correct tree always total 1. When items are not put back, the second-stage branches carry new numbers.
How to spot it
- An experiment runs in stages, such as pulling two tokens from a bag one after another.
- The wording says without replacement, so the second set of chances differs from the first.
- You need to be sure that no combination of results gets skipped.
- Several different routes can produce the event, and their chances must be added.
The method
- Sketch a branch for every possible first result and label each with its probability.
- From the tip of each branch, sketch the second-stage branches with updated probabilities.
- Multiply the numbers along each route to get the chance of that outcome.
- Add the routes that match the event, then verify that all tips sum to 1.
Tree Diagram: solved example
Problem
A pouch holds 4 green and 3 yellow tokens. Two tokens are drawn one at a time, and the first is not returned. Use a tree to find the probability that both are green.
Solution
First draw green: \(\frac{4}{7}\). One green is gone, so 3 greens remain among 6 tokens: second draw green is \(\frac{3}{6}\).
Multiply along the route: \(\frac{4}{7} \times \frac{3}{6} = \frac{12}{42} = \frac{2}{7}\). The probability is \(\frac{2}{7}\).
Try it
Question 1
With the same pouch, how likely is it that the two tokens differ in color?
Show the answer
The routes GY and YG each carry \(\frac{12}{42}\), so \(\frac{12}{42} + \frac{12}{42} = \frac{24}{42} = \frac{4}{7}\).
Question 2
A coin flip is followed by a spinner with 3 equal sections. How many tips does the tree have?
Show the answer
The coin gives 2 branches and each leads to 3 spinner branches, so \(2 \times 3 = 6\) tips.
Tree Diagram vs. Sample Space
The sample space is the list of all outcomes, and it can be shown as a set, a table or an organized list. A tree diagram is one drawing that produces that list, with the bonus of printing chances on its branches, which is handy when the second stage depends on the first.
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Whole dictionary: A–Z · 7th Grade · Probability