Sequences, Probability and Counting Theory

A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a new home? A luxury convertible? A cruise around the world? The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over an extended period of time. This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum. In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning.

Sections

• Sequences and Their Notations — write the terms of a sequence defined by an explicit formula or by a recursive formula, find explicit formulas from the first few terms, and use factorial notation.
• Arithmetic Sequences — find the common difference of an arithmetic sequence, write its terms, and use recursive and explicit formulas for its $n$th term, including in applications.
• Geometric Sequences — find the common ratio of a geometric sequence, list its terms, and use recursive and explicit formulas for its $n$th term.
• Series and Their Notations — use summation notation and the formulas for the sum of the first $n$ terms of an arithmetic or geometric series and for the sum of an infinite geometric series, and solve annuity problems.
• Counting Principles — solve counting problems using the Addition and Multiplication Principles, permutations of distinct and non-distinct objects, combinations, and the number of subsets of a set.
• Binomial Theorem — find binomial coefficients, use the Binomial Theorem to expand a binomial, and find a single term of a binomial expansion.
• Probability — construct probability models, compute probabilities of equally likely outcomes and of the union of two events, use the Complement Rule, and compute probabilities with counting theory.

The sections above are adapted from Precalculus 2e, Chapter 11 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. The chapter opener’s lottery-ticket photograph (credit: Robert Couse-Baker, Flickr) is omitted; its introduction is reproduced above.