This lesson deals with both probability and geometric series. Probability is usually introduced as applying to a single event (such as randomly choosing from a set of colored balls), but in this case, because victory can be achieved in infinitely many different ways, the desired probability is the sum of infinitely many probabilities.

The concept of an infinite series with a finite sum is often introduced in second-semester calculus courses. One strategy to estimate the value of an integral of a function (when the function has a difficult or non-elementary antiderivative) is to write the function as an infinite sum of polynomial terms, which is called a Taylor series. If the Taylor series has a finite sum, that sum is (usually) equal to the value of the integral. In practice, you can add up some of the terms to get a good estimate for this integral value.

In this lesson, even though the probability we want is the sum of infinitely many probabilities, we are fortunate that those probabilities fit the pattern of a geometric series. Unlike most infinite series, series with this special geometric structure are predictable, in terms of whether they converge and what they converge to. This allows us to find the sum of infinitely many probabilities using a single formula.

Adding up an infinite series that is not geometric can be much more complicated. Sometimes this goal can be achieved by writing the series as a sum of many geometric series, or by using advanced tools such as generating functions. But often the exact sum is difficult or impossible to find, which is why the integration strategy described above is often used to find estimates rather than exact values.

Next Section »