Probability Tree Diagrams — How to Draw & Calculate

A probability tree diagram is a picture of a chance process that unfolds in stages. You draw one branch for each possible outcome, label each branch with its probability, multiply along a full path, and add the probabilities of different paths that all produce the event. It earns its keep when later probabilities depend on earlier results, because then the numbers on later branches change from path to path and the tree keeps those conditions visible.

The Two Rules That Do the Work

A tree starts at one point and branches outward. The first branches show the first stage; the next branches show what can happen after each earlier outcome. Each complete path is one full scenario — in a two-draw bag problem, the path $RB$ means "red first, then blue."

Almost all the calculation comes from two rules.

Multiply along one path:

If the second stage does not depend on the first, then $P(B \mid A) = P(B)$ and the multiplication is simpler. You multiply because you want several events to happen in sequence along a single path.

Add across paths: if the event can happen through more than one successful path, add those path probabilities. You add because the paths are different, mutually exclusive ways for the same event to occur.

Drawing the Tree, Step by Step

1. Name the stages clearly — a first draw and a second draw, a first toss and a second toss.
2. From each node, draw every possible outcome for that stage.
3. Label each branch with the probability that applies at that node. This is the step students rush: under "without replacement" or with extra information, later branch probabilities change.
4. Check each node: the branches leaving the same node should add to $1$. If they do not, the tree is incomplete or a probability is wrong.

Full Worked Example: Two Draws Without Replacement

A bag holds $2$ red balls and $1$ blue ball. You draw one ball, do not replace it, then draw a second. What is the probability of exactly one red and one blue?

First draw:

• Red: $P(R) = \frac{2}{3}$
• Blue: $P(B) = \frac{1}{3}$

Now update the second-draw probabilities on each branch. If the first ball is red, the bag has $1$ red and $1$ blue left:

• $P(R \mid R) = \frac{1}{2}$
• $P(B \mid R) = \frac{1}{2}$

If the first ball is blue, the bag has $2$ red and $0$ blue left:

• $P(R \mid B) = 1$
• $P(B \mid B) = 0$

Multiply along each complete path:

"Exactly one red and one blue" happens through two paths, $RB$ or $BR$, so add them:

This is exactly why trees help: the second-draw probabilities are not fixed — they depend on the first draw, and the tree makes that dependence easy to see.

Practice With Your Own Bag

Set up a bag with $3$ green balls and $2$ yellow balls. Draw two without replacement and find the probability that both balls are the same color. Build the tree, update the second-draw branches, confirm each node sums to $1$, then multiply along paths and add. Comparing your tree against a step-by-step solution afterward is a fast way to catch a missed branch.

Calculation Traps to Avoid

• Forgetting to update later probabilities. Without replacement, or after learning new information, later branches change. Reusing the starting probability everywhere gives the wrong tree.
• Adding when you should multiply. Along a single path you want several things to happen in sequence, so multiply.
• Multiplying when you should add. Across different successful paths such as $RB$ or $BR$, compute each and then add.
• Leaving out an impossible branch. A probability-$0$ branch still belongs on the tree, because it shows the outcome cannot happen from that point.