Probability Tree Diagram

What is a probability tree?

A probability tree diagram maps out all possible outcomes of sequential random events, with probabilities written on each branch. It's the most visual way to organize multi-step probability problems.

Two key rules

• Multiplication rule (along a path): Multiply probabilities along each branch to find the probability of that specific path.
• Addition rule (across paths): Add the probabilities of all paths that lead to the desired event.

Step-by-step: Two fair coin flips

Step 1 — Draw the first flip. Two branches: H (probability 0.5) and T (0.5).

Step 2 — From each outcome, draw the second flip. Each splits into H (0.5) and T (0.5).

Step 3 — List all paths and their probabilities:

• HH: $0.5 \times 0.5 = 0.25$
• HT: $0.5 \times 0.5 = 0.25$
• TH: $0.5 \times 0.5 = 0.25$
• TT: $0.5 \times 0.5 = 0.25$

Total: $0.25 \times 4 = 1.00$ ✓ (all probabilities sum to 1).

Step 4 — Answer questions using the tree.

$P(\text{at least one H})$ = HH + HT + TH = $0.25 + 0.25 + 0.25 = 0.75$.

Or: $P(\text{at least one H}) = 1 - P(\text{no H}) = 1 - P(\text{TT}) = 1 - 0.25 = 0.75$.

When the coin is unfair

If $P(H) = 0.7$, the branches have different probabilities. HH = $0.7 \times 0.7 = 0.49$, TT = $0.3 \times 0.3 = 0.09$. The tree makes this easy to track.

Try it in the visualization

Adjust the probability of heads. The tree branches with probabilities on each edge. Outcomes and their probabilities update live. Highlight specific events to sum their probabilities.