Probability Tree Diagram Formula & How to Use It

This calculator works out probabilities along a probability tree diagram. You enter the probability of each branch, and it returns the combined probability for the path you select, along with the complementary probability. The formula is simply the product of the probabilities along the branches of the path you are interested in.

Formula

The probability of a specific path through a tree diagram is the product of the probabilities of each branch along that path. In symbols:

P(path) = P(branch 1) × P(branch 2) × … × P(branch n)

Where each P(branch) is the probability written on that branch (a number between 0 and 1). For example, if a path has three branches with probabilities 0.6, 0.7, and 0.4, then P(path) = 0.6 × 0.7 × 0.4 = 0.168 (16.8%).

Worked Example

Using the default values from the calculator: p1 = 0.5 and p2 = 0.5.

1. Multiply the branch probabilities: 0.5 × 0.5 = 0.25.
2. The probability of this path is 0.25, or 25%.
3. The complement (probability the path does not happen) is 1 − 0.25 = 0.75, or 75%.

The calculator shows the same result: path probability 0.2500 and complement 0.7500.

When to Use a Probability Tree

Probability trees are useful for problems with two or more sequential events, especially when the events are independent (the outcome of one does not affect the next) or when you need to consider conditional probabilities. Common examples include:

• Flipping a coin twice and finding the probability of getting heads both times.
• Drawing two marbles from a bag without replacement (the second probability changes).
• Testing a product that passes or fails two quality checks in a row.
• Calculating the chance of a medical test being correct given a disease prevalence (though that typically uses Bayes’ theorem).

If the events are independent, the tree is straightforward. If they are dependent, the branch probabilities must reflect the changed conditions (e.g., after drawing one marble, the total count changes).

How to Read the Result

The result is the probability of that one specific sequence of outcomes. To get the probability of any of several paths (e.g., “at least one head” in two flips), you add the probabilities of all the paths that satisfy the condition. For two coin flips, the paths are HH, HT, TH, TT, each with probability 0.25. The probability of at least one head is 0.25 + 0.25 + 0.25 = 0.75.

The complement (1 − P(path)) is the probability that the exact sequence does not occur. That is not the same as “the opposite outcome” unless the path is the only way to get that outcome. Always check what the complement actually represents in your problem.

Common Mistakes

• Forgetting to multiply all branches – skipping a branch gives a wrong, usually too large, probability.
• Using percentages instead of decimals – 50% must be entered as 0.5, not 50. The formula multiplies decimals.
• Adding instead of multiplying – multiply along a path, add across different paths that satisfy the same condition.
• Ignoring conditional changes – when drawing without replacement, the probability on the second branch changes after the first draw. For example, drawing two red marbles from a bag of 3 red and 2 blue: first draw P(red) = 3/5 = 0.6, but second draw P(red) = 2/4 = 0.5, so P(both red) = 0.6 × 0.5 = 0.3.

Limits of the Formula

The multiplication rule works for any finite number of branches, but it assumes the probabilities on each branch are correct for the situation. If the events are not independent and you do not adjust the branch probabilities, the result will be wrong. Also, the tree method becomes unwieldy for many events (e.g., 10 coin flips would have 2¹⁰ = 1024 paths). In those cases, use the binomial distribution or other methods. The formula also assumes the tree is exhaustive – every possible outcome is represented by some path, and the probabilities on all branches from any node sum to 1.

Frequently Asked Questions

Why do we multiply probabilities along branches?

Because each branch represents a condition that must happen for the path to occur. The probability of two independent events both happening is the product of their individual probabilities. This is the multiplication rule of probability.

What if the branches are not independent?

You must use conditional probabilities. For example, if you draw two cards without replacement, the probability of the second card being red depends on whether the first was red. Write the conditional probability on the second branch (e.g., P(red | first red) = 25/51). The multiplication rule still works, but the branch values must be conditional.

How do I find the probability of “at least one” using a tree?

Find the probability of the complementary event (none of the outcomes happen) and subtract from 1. For example, the probability of at least one head in two flips is 1 − P(TT) = 1 − 0.25 = 0.75. This is often faster than adding all the favorable paths.

Can I use a tree for more than two events?

Yes. Trees can have any number of levels. For three coin flips, you would have 2³ = 8 paths. The formula is the same: multiply the probabilities along the path. But the tree grows exponentially, so for more than about 5–6 events, other methods are usually easier.

What is the difference between a tree diagram and a Venn diagram?

A tree diagram shows sequential events and their probabilities along branches. A Venn diagram shows sets and their overlaps for a single event or a few events. Trees are better for multi-step processes; Venn diagrams are better for visualizing intersections and unions of sets.

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in p1 | Probability of first branch (0-1) | - | 0.5 | 0
in p2 | Probability of second branch (0-1) | - | 0.5 | 0
out path | Probability of this path | - | p1 * p2 | 4
out complement | Probability of not this path | - | 1 - p1 * p2 | 4