### Topic Page: tree diagram

**tree diagram**from

*The Hutchinson Unabridged Encyclopedia with Atlas and Weather Guide*

In probability theory, a branching diagram consisting only of arcs and nodes (but not loops curving back on themselves), which is used to establish probabilities. Each branch represents a possible outcome of one event. These possible outcomes are known as mutually exclusive events. The final probability depends on the path taken through the tree.

For example, when two cakes are baked the probability that one cake will not be burnt is 0.95. The tree diagram shows all the possible outcomes:

The outcome of each cake can be described as an independent event as they have no effect on each other. Each path through the tree gives a different outcome. In this case there are four outcomes.

The dotted path gives the outcome of both cakes burning. The probabilities are multiplied to obtain this outcome:

Probability of both cakes burning = 0.05 × 0.05 = 0.0025

The dashed path gives the outcome of both cakes not burning:

Probability of both cakes not burning = 0.95 × 0.95 = 0.9025

The probability of one cake burning has two outcomes: (i) the first burnt and second not burnt, and (ii) the first not burnt and the second burnt. These probabilities are:

(i) 0.05 × 0.95 = 0.0475 and (ii) 0.95 × 0.05 = 0.0475

The two probabilities are added together to obtain the probability of one cake burning = 0.0475 + 0.0475 = 0.095.

The four probabilities should add up to 1.

Probabilities of independent events are always multiplied, characterized by and, for example, first cake burnt and second cake burnt. Probabilities of mutually exclusive events are always added, characterized by or, for example, first cake burnt or not burnt.

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Probability calculation using a tree diagram

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