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Tree Diagram

A tree diagram is a branching diagram that maps out all possible outcomes of a sequence of events. Each branch represents one possible result at each stage, and following a path from start to finish gives you one complete outcome.

A tree diagram is a graphical representation used in probability to display every possible outcome of two or more sequential events. Each node in the diagram branches into the possible results of that event, and the total number of paths from root to endpoints equals the size of the sample space. When probabilities are assigned to each branch, the probability of any complete outcome is found by multiplying the probabilities along its path.

Key Formula

P(outcome)=P(branch1)×P(branch2)×⋯×P(branchn)P(\text{outcome}) = P(\text{branch}_1) \times P(\text{branch}_2) \times \cdots \times P(\text{branch}_n)
Where:
  • P(outcome)P(\text{outcome}) = the probability of one complete path through the tree
  • P(branchk)P(\text{branch}_k) = the probability assigned to the branch at stage k

Worked Example

Problem: You flip a coin and then roll a six-sided die. Use a tree diagram to find the probability of getting Heads and an even number.
Step 1: Draw the first set of branches for the coin flip. There are two outcomes: Heads (H) and Tails (T), each with probability 1/2.
P(H)=12,P(T)=12P(H) = \frac{1}{2}, \quad P(T) = \frac{1}{2}
Step 2: From each coin outcome, draw six branches for the die roll (1 through 6), each with probability 1/6. This gives 2 × 6 = 12 total paths.
Total outcomes=2×6=12\text{Total outcomes} = 2 \times 6 = 12
Step 3: Identify the paths that match "Heads and an even number." These are H-2, H-4, and H-6 — three paths out of twelve.
Step 4: Multiply the probabilities along one such path to find its probability, then add the three matching paths together.
P(H and even)=3×(12×16)=3×112=312=14P(H \text{ and even}) = 3 \times \left(\frac{1}{2} \times \frac{1}{6}\right) = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}
Answer: The probability of getting Heads and an even number is 14\frac{1}{4}, or 0.25.

Why It Matters

Tree diagrams are one of the most practical tools for solving probability problems because they force you to list every outcome systematically, making it hard to miss any. They appear throughout statistics, genetics (predicting trait inheritance), sports brackets, and decision-making models. Whenever a problem involves a sequence of choices or events, a tree diagram can help you organize and count possibilities clearly.

Common Mistakes

Mistake: Adding branch probabilities along a path instead of multiplying them.
Correction: To find the probability of a complete outcome, you multiply the probabilities along the path. Addition is used when you want to combine the probabilities of separate paths (different outcomes).
Mistake: Forgetting to include all branches at each stage, which leads to a missing outcome.
Correction: At every node, make sure you draw one branch for each possible result of that event. The total number of endpoints should equal the product of the number of options at each stage.

Related Terms

  • Probability — Tree diagrams are used to calculate probabilities
  • Sample Space — A tree diagram displays the entire sample space
  • Outcome — Each path through the tree represents one outcome