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Push the frontiers of your knowledge with these probabilistic techniques.

These quizzes explore the advanced techniques that can be used to calculate probabilities. Many of these techniques are useful when the number of "successes" and total outcomes are especially complicated to compute.

Geometric probability is a tool to deal with the problem of infinite outcomes by measuring the number of outcomes geometrically, in terms of length, area, or volume. Dealing with continuous variables can be tricky, but geometric probability provides a useful approach by allowing us to transform probability problems into geometry problems.

Often, rather than directly counting the number of ways to do something, an equivalent set of objects that is easier to count can be found, and that set counted instead.

For example, consider the problem of distributing 5 indistinguishable balls into 3 distinguishable urns. This is hard to directly count, but if we instead imagine 5 balls and two dividers, it is sufficient to count the number of ways to permute a string of 5 balls and 2 dividers, since each permutation will give rise to an arrangement of balls into urns.

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