Tree diagrams, selection with/without replacement, dependence
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Practice exam-style dependence problems
A tree diagram visually represents possible outcomes of multiple-step probabilistic events. Each branch splits to represent different outcomes of each stage, with probabilities written along the branches. Tree diagrams simplify complex probability problems, clearly showing how probabilities combine at each step.
To find the probability of a particular combination of events -- a node, or "leaf," on the far right side of the tree diagram -- multiply probabilities along its branches. To find the probability of an event with multiple outcomes, sum the probabilities of each relevant branch.
For example, a tree diagram portraying the outcomes of an experiment with two trials, where the outcomes are two complementary events, looks like:
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In probability, a selection is the action of choosing one or more items from a set or group. We can perform selections either with replacement or without replacement.
In selection with replacement, each chosen item is returned to the original group before the next choice, keeping the probabilities constant across selections.
In selection without replacement, the chosen items are removed from the group, causing probabilities to change after each pick because the number of available items decreases.
This difference significantly impacts how probabilities are calculated, especially in problems involving multiple selections.