Introduction to Probability

How do you use a tree diagram for a two stage experiment?

Quick answer

To use a tree diagram for a two stage experiment, draw branches from a starting point for each outcome of the first stage, then draw new branches from each tip for the second stage. Label each branch with its probability. Multiply the probabilities along a single path to find an outcome, and add the results of different paths together.

The topic explained

What is a tree diagram for a two stage experiment?

A tree diagram for a two stage experiment organizes all possible outcomes in order. The diagram begins at a single starting point on the left. Branches spread out for every possible outcome of the first stage. From each of those endpoints, new branches spread out for the outcomes of the second stage. Every complete path from left to right represents one specific compound outcome.

Write the probability of each event directly on its branch. At any single node, the probabilities on all branches leaving it must add up to 1. This quick check confirms that no possibilities were missed.

Multiply along paths, add between paths

Finding an overall probability requires two clear rules. First, multiply along a path. A complete path represents an intersection where stage one and stage two both occur. For example, if drawing a red marble is 3 out of 5 on the first branch and 2 out of 4 on the second branch, multiply 3/5 by 2/4 to get 6/20, which simplifies to 3/10.

Second, add between paths. If an event can happen in multiple ways—such as picking one red and one blue marble in any order—identify every path that meets the condition and add their products together. Never add probabilities along a single path.

Sampling with replacement versus without replacement

Pay close attention to whether items are replaced. In sampling with replacement, the item is returned before the second stage. The total number of items stays the same, so the denominator and probabilities on the second-stage branches remain unchanged.

In sampling without replacement, the item is kept. The total count drops by one, so every second-stage denominator decreases by one. The numerator decreases only for the specific item that was just selected. If a math problem states that two items are drawn together, treat it as sampling without replacement.

Free printable worksheet

Two-Stage Probability Tree Diagrams Worksheet

For 10th grade students. Calculate the probability for each two-stage scenario using tree diagram rules. Simplify fractions where possible.

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Two-Stage Probability Tree Diagrams WorksheetName:

A coin is flipped twice. What is the probability of getting Heads on both flips?
A bag holds 3 red marbles and 2 blue marbles. One marble is drawn, noted, and replaced before drawing a second. What is the probability of drawing two red marbles?
From the same bag (3 red, 2 blue), two marbles are drawn without replacement. What is the probability that both marbles are red?
Two marbles are drawn without replacement from a bag of 3 red and 2 blue marbles. What is the probability of drawing one red and one blue marble in any order?
A spinner has 3 equal sections: Red, Green, and Yellow. It is spun twice. What is the probability of spinning Green at least once?
A drawer contains 4 black socks and 2 white socks. You take two socks without replacement in the dark. What is the probability of picking a matching pair?

Common questions

Why do we multiply along the branches instead of adding?

A path along the branches represents an intersection where stage one and stage two both occur. In probability, finding the chance of consecutive events happening requires multiplying their probabilities. Adding along a path would produce an incorrect result, often greater than 1.

What is the most common mistake made on tree diagram problems?

The most common mistake is forgetting that an outcome can happen in more than one order. For example, obtaining one red and one blue item involves two distinct paths: Red-then-Blue and Blue-then-Red. Calculating only one path gives exactly half the true probability.

How do you know when to change the denominator on the second branch?

Check if items are replaced. If an item is put back, the denominator stays the same. If an item is removed, or if items are taken simultaneously, the total number decreases, so the denominator drops by one for the second stage.

How can you check if your tree diagram is drawn correctly?

Check the branches originating from each individual node; their probabilities must always add up to 1. In addition, the final calculated probabilities of every end branch across the whole tree must add up to 1.

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