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Mathematics › 🔄 Grade 7 › Tree Diagrams and the Probability of Two Events Together
🔄 Grade 7 · Lesson 12 of 14

Tree Diagrams and the Probability of Two Events Together

HT and TH look the same — but they're two different paths. The tree shows you why.

Grade 7Middle School
Tree Diagrams and the Probability of Two Events Together — illustration
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The big idea: When two independent events happen together (a coin flip then a die roll, for example), the full set of outcomes is called the sample space. A tree diagram builds this space one branch at a time and makes it impossible to miss any outcome. The probability of any specific combined outcome is the product of the probabilities along its branch.
🎯 By the end, you'll be able to
  • List the complete sample space for two sequential independent events using a tree diagram
  • Recognize that HT and TH are different outcomes even though they describe the same number of heads
  • Calculate the probability of a specific combined outcome by multiplying branch probabilities
  • Use a two-way table (grid) as an alternative representation of the sample space
📎 You should already know
  • Single-event experimental vs theoretical probability
  • Simple fractions and their multiplication

The classic trap: 'two heads,' 'one head,' 'no heads' aren't equally likely!

Flip two coins. Most students say: “There are three outcomes — 0 heads, 1 head, 2 heads — so each has probability 1/3.” But that is wrong.

Draw a tree and look carefully at “1 head.” You can get it as HT (first coin heads, second tails) or as TH (first tails, second heads). Those are two separate paths, not one. The actual sample space has 4 equally-likely outcomes: HH, HT, TH, TT. So P(exactly 1 head) = 2/4 = 1/2, not 1/3.

Building a tree diagram one fork at a time

Start with a single dot (the “start” node). Draw one branch for each outcome of Event 1. At the end of each branch, draw one sub-branch for each outcome of Event 2. Each leaf at the end is one complete outcome of the combined experiment.

  • Number of leaves = (outcomes of Event 1) × (outcomes of Event 2).
  • Each leaf's probability = probability of the path from root to that leaf = P(Event 1 outcome) × P(Event 2 outcome).
  • All leaf probabilities add up to 1.
🎮 Compound Probability Tree LIVE
Select Event 1 and Event 2 (coin or die). Each leaf shows one combined outcome. Highlighted leaves show a target event. Toggle Tree ↔ Grid to see the sample space as a two-way table.
🔑 Multiplying along a branch

For two fair coins, each flip has P(H) = 1/2 and P(T) = 1/2. The probability of landing on the HH leaf is:

P(H) × P(H) = 1/2 × 1/2 = 1/4

In general: for independent events, P(A and B) = P(A) × P(B).

This is why the tree is so powerful: multiply along the branch, and you get the exact probability of that combined outcome without needing any formula.

📝 Worked example: Flip a fair coin and roll a fair 4-sided die (faces 1–4). What is P(Heads and a number less than 3)?
  1. Event 1 (coin): P(H) = 1/2.
  2. Event 2 (die < 3): outcomes that work are {1, 2} out of {1,2,3,4}, so P(<3) = 2/4 = 1/2.
  3. Multiply: P(H and <3) = 1/2 × 1/2 = 1/4.
  4. Verify with the tree: total leaves = 2 × 4 = 8 equally likely. Matching leaves: H1 and H2 → 2 leaves. P = 2/8 = 1/4. ✓
✓ P(H and &lt; 3) = <strong>1/4</strong>.
✨ Grid view = same information, different layout

A two-way table (grid) puts Event 1 outcomes across the top and Event 2 outcomes down the side. Each cell is one combined outcome. Shading the cells that match your target event and counting them gives the same probability as the tree diagram.

Some people find the grid easier to read for small sample spaces; the tree diagram is clearer when the two events have different numbers of outcomes.

Check your understanding

1. Flip two fair coins. What is P(exactly 1 head)?
Sample space: HH, HT, TH, TT — 4 equally likely outcomes. Exactly 1 head: HT and TH — 2 outcomes. P = 2/4 = 1/2.
2. Flip a fair coin and roll a fair 6-sided die. How many outcomes are in the sample space?
2 coin outcomes × 6 die outcomes = 12 leaves in the tree.
3. What is P(Tails and a 3) when you flip a coin and roll a fair 6-sided die?
P(T) = 1/2, P(3) = 1/6. Multiply: 1/2 × 1/6 = 1/12.
4. Why is P(exactly 1 head in two coin flips) = 1/2 and NOT 1/3?
The sample space has 4 outcomes (HH, HT, TH, TT). 'Exactly 1 head' corresponds to 2 of them (HT and TH), not 1. So P = 2/4 = 1/2.
5. Flip a coin and roll a 6-sided die. What is P(Tails and a number less than 3)?
P(T) = 1/2. P(die < 3) = 2/6 = 1/3. P(T and <3) = 1/2 × 1/3 = 1/6. Check: 2 matching leaves (T1, T2) out of 12 total = 2/12 = 1/6. ✓
✅ Key takeaways
  • A tree diagram lists every possible combined outcome of two sequential events — no outcome gets missed.
  • The sample space size = (outcomes of Event 1) × (outcomes of Event 2).
  • HT and TH are distinct outcomes; always count them separately.
  • Probability of any leaf = P(Event 1 outcome) × P(Event 2 outcome) — multiply along the branch.
  • A two-way table (grid) shows the same sample space in a different format — same probabilities result.
  • The equiprobability fallacy: 0, 1, and 2 heads are NOT equally likely for two coin flips.