In a typical \(6/49\) lotto, where \(6\) numbers are drawn from a range of \(49\) and if the six numbers drawn match on the ticket, the ticket holder is a jackpot winner. The odds of this event are \(1\) in \(13,983,816\).
The study of counting methods dates at least to Gottfried Wilhelm Leibniz’s Dissertatio de Arte Combinatoria in the seventeenth century. Combinatorics has applications in many fields of study. Applications of combinatorics arise, for example in chemistry, in studying the arrangements of atoms in molecules and crystals; biology in questions about the structure of genes and proteins; physics, in problems in statistical mechanics; communication, in the design of codes for encryption, compression, and especially in computer science, for instance in problems of scheduling and allocating resources.
The links to my notes on how to count and the solved problems are given below.
Theorems and notes
Exercises and examples
More solved problems
The points of a sample space may be very concrete objects. For example, on tossing a coin, a head or tail is realized, on tossing a coin twice, a head-head, head-tail, tail-head or tail-tail is realized, on rolling die, one of the numbers \(1,2,3,4,5,6\) is realized, on drawing a card from a standard deck of cards, one of four suits – club, spade, heart, diamond and one of \(13\) ranks is realized.
However, sometimes the sample space of an experiment can be incredibly complicated or higher dimensional and the outcomes may be non-numeric. For example, in a sequence of \(N\) coin tosses, the sample space has an enormous \(2^N\) sample points. In a survey of a random sample of people in a city, various questions may have numeric (e.g. age or height) and non-numeric answers (e.g. favorite political party or favorite movie). Thus, we require a mathematical description of the sample points.
Random variables fulfill this purpose. They assign numerical values to each sample point.
Random variables and their distributions