Probability Tree Diagrams — How to Draw & Calculate

A probability tree diagram is a picture of a chance process that happens in stages. You draw one branch for each possible outcome, label each branch with its probability, multiply along a full path, and add different successful paths when the event can happen in more than one way.

It is most useful when later probabilities depend on earlier results. In that case, the probabilities on later branches may change from one path to another, so the tree helps you keep the conditions visible.

What probability tree diagrams show

A probability tree starts from one point and branches outward. The first branches show the first stage of the experiment, and the next branches show what can happen after each earlier outcome.

Each complete path represents one full scenario. If a bag problem has two draws, then a path such as $RB$ means "red first, then blue."

Two rules do almost all of the calculation:

$$P(A \text{ then } B) = P(A)P(B \mid A)$$

Use this rule for one full path. If the second stage does not depend on the first, then $P(B \mid A) = P(B)$, so the multiplication is simpler.

If the event can happen through more than one successful path, add those path probabilities.

How to draw a probability tree diagram

Start by naming the stages clearly. For example, you might have a first draw and a second draw, or a first toss and a second toss.

From each node, draw every possible outcome for that stage and label each branch with the probability that applies at that node. This is the part students often rush. If the setup says "without replacement" or gives extra information, later branch probabilities can change.

One quick check catches many mistakes: the branches leaving the same node should add to $1$. If they do not, the tree is incomplete or one of the probabilities is wrong.

When to multiply and when to add

Multiply when you stay on one path and want the probability that several events happen in sequence.

Add when the final event can happen through different complete paths. For example, "exactly one red and one blue" can happen as $RB$ or $BR$, so you calculate each path first and then add them.

Probability tree diagram example: two draws without replacement

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

Start the tree with the first draw:

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

Now update the second-draw probabilities on each branch.

If the first ball is red, the bag has $1$ red and $1$ blue left, so:

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

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

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

Now multiply along each complete path:

$$P(RR) = \frac{2}{3}\cdot\frac{1}{2} = \frac{1}{3}$$ $$P(RB) = \frac{2}{3}\cdot\frac{1}{2} = \frac{1}{3}$$ $$P(BR) = \frac{1}{3}\cdot 1 = \frac{1}{3}$$ $$P(BB) = \frac{1}{3}\cdot 0 = 0$$

"Exactly one red and one blue" happens in two different paths: $RB$ or $BR$. Add those two path probabilities:

$$P(\text{exactly one red and one blue}) = P(RB) + P(BR) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$$

This is the main reason tree diagrams help so much. The second-draw probabilities are not fixed; they depend on the first draw, and the tree makes that dependence easy to see.

Common mistakes with probability trees

Forgetting to update later probabilities

If the experiment is without replacement, or if you learn new information after the first stage, the next probabilities may change. Reusing the starting probability at every branch gives the wrong tree.

Adding when you should multiply

Along a single path, you are finding the probability that several things happen in sequence, so you multiply.

Multiplying when you should add

If an event can happen through more than one successful path, such as $RB$ or $BR$, calculate each path and then add them.

Leaving out an impossible branch

Sometimes a branch has probability $0$. It still matters because it shows that the outcome cannot happen from that point.

When probability tree diagrams are used

Tree diagrams are common in basic probability, card and ball-drawing problems, medical testing with staged outcomes, and any setup where events happen in order.

They are also a good bridge into conditional probability. Even if you later switch to formulas, the tree is often the fastest way to see the structure of the problem first.

Try your own version

Try a similar problem with a bag that contains $3$ green balls and $2$ yellow balls. Draw two balls without replacement and find the probability that both balls are the same color.

If you want a next step after that, try your own version in GPAI Solver and compare your tree with a step-by-step solution.