Probabilities and Expectations

Abstract

Probabilities tell us about the likelihood of an event in numbers. If an event is certain to occur, such as sunrise, probability of that event is said to be 1. Pr(sunrise) = 1. If an event will certainly not occur, then its probability is 0. So, probability maps events to a number in [0, 1]. How do you specify an event? In the discussions of probabilities, events are technically described as a set. At this point it is important to go through some basic concepts of sets and maybe also functions. Sets A set is a collection of distinct objects. For example, if we toss a coin once, the set of all possible distinct outcomes will be S = {head, tail}, where head denotes a head and the tail denotes a tail. All sets we consider here are finite. An element of a set is denoted as head ∈ S. A subset of a set is denoted as {head} ⊂ S. What are the possible subsets of S? These are: {head} , {tail} , S = {head, tail}, and φ = {}. So, note that a set is a subset of itself: S ⊂ S. Also note that, an empty set (a collection of nothing) is a subset of any set: φ ⊂ S. A union of two sets A and B is comprised of all the elements of both sets and denoted as A ∪ B. An intersection of two sets A and B is comprised of only the common elements of both sets and denoted as A ∩ B. A complement set of A in B is a set comprising the elements of B that are not in A and denoted as B \ A or B − A. The Cartesian product of two sets A and B is a set denoted as A × B comprising all ordered pairs (a, b) where a ∈ A and b ∈ B.