Compound Probability Using Tree Diagrams- Examples and Practice

What Compound Probability Actually Is

Compound probability is the likelihood of two or more events happening together. That's it. Nothing fancy.

There are two types you need to know:

Most students struggle with compound probability because they try to memorize formulas instead of visualizing what's happening. That's where tree diagrams come in.

Why Tree Diagrams Work

Tree diagrams show every possible outcome at each step. You can see the full picture instead of guessing.

Each branch represents one outcome. You multiply along the branches to find compound probabilities. You add probabilities only when paths are separate (either this OR that happens).

How to Draw a Tree Diagram

Step 1: Start with the first event

Draw a dot on the left. Branch out for each possible outcome of the first event. Write the probability on each branch.

Step 2: Add the second event

From each first-event outcome, draw branches for the second event. Write those probabilities.

Step 3: Continue until done

Keep going for however many events you're analyzing.

Step 4: Calculate

For any single path: multiply all probabilities along that path. For multiple paths: add the individual path probabilities.

Example 1: Two Coin Flips (Independent Events)

What's the probability of flipping exactly one head in two coin tosses?

Step 1: First flip — Heads (0.5) or Tails (0.5)

Step 2: Second flip from each outcome — Heads (0.5) or Tails (0.5)

Here's the tree:

First Flip → Second Flip → Path Probability

Exactly one head means either Heads-Tails or Tails-Heads:

P = 0.25 + 0.25 = 0.5 (or 50%)

Example 2: Drawing Marbles (Dependent Events)

A bag has 3 red and 2 blue marbles. You draw two marbles without replacement. What's P(Red, then Blue)?

This is where most people mess up — they use the same probability twice.

First draw: P(Red) = 3/5

Second draw: After removing a red marble, you have 4 marbles left (2 red, 2 blue). P(Blue) = 2/4 = 1/2

Tree diagram shows:

Check: 3/10 + 3/10 + 3/10 + 1/10 = 10/10 = 1 ✓

Example 3: Three Events

You roll a die three times. What's P(even, odd, even)?

Each roll is independent. P(even on one roll) = 3/6 = 1/2. P(odd) = 1/2.

Path: Even × Odd × Even = 1/2 × 1/2 × 1/2 = 1/8

Simple when you map it out.

Tree Diagrams vs. Other Methods

Here's how tree diagrams compare to other approaches:

Method Best For Drawback
Tree Diagrams 2-3 events, dependent events, visual learners Gets messy with many events
Probability Formulas Quick calculations, independent events Easy to use wrong formula
Counting Outcomes Simple equally-likely events Falls apart with weighted probabilities
Tables Two-variable problems Doesn't extend to three+ events

Practice Problems

Problem 1: A drawer has 4 black socks and 2 white socks. You grab two socks (no replacement). What's the probability both are black?

Answer: P(Black, Black) = 4/6 × 3/5 = 12/30 = 2/5 = 0.4

Problem 2: You flip a coin three times. What's P(exactly 2 heads)?

Answer: Paths with 2 heads = HHT, HTH, THH. Each = 1/8. P = 3/8 = 0.375

Problem 3: A basket has 2 apples (1 rotten) and 3 oranges (1 rotten). You pick two fruits. What's P(both are good)?

Answer: P(Apple good) = 1/2, P(Orange good) = 2/3. Need to consider order: Apple then Orange OR Orange then Apple = (1/2 × 2/3) + (3/5 × 1/2) = 2/6 + 3/10 = 20/60 + 18/60 = 38/60 = 19/30 ≈ 0.633

Common Mistakes

Quick Reference

Remember these rules:

Tree diagrams aren't magic. They're just a way to organize your thinking so you don't miss any outcomes or use the wrong operation. Draw them out, label everything, and your probability problems solve themselves.