ABSTRACT
A probability function, P (A), is a function from (certain) subsets of a space S to the interval [0, 1]. (We will not worry about which subsets-all “reasonable” subsets are included.) The space S is referred to as the sample space, and the permitted subsets of S are often called events. Probability functions must satisfy the following three rules:
1. P (φ) = 0
2. P (A ∪B) = P (A) + P (B) if A ∩B = φ 3. P (S) = 1
where φ = {}—φ is the empty set and A and B are events. Two events are said to be independent if:
P (A ∩B) = P (A)P (B). (1.1)
The theory that we develop in this book is a mathematical theory. It is, however, supposed to reflect physical reality. A probability function is a mathematical function with certain properties. It is supposed to be related to the “actual probability” that some event will occur. This means that when we choose the probability function we try to make sure that if P (A) = 0, then
α, out trials we expect A to occur αN times. Let us consider a simple example of an attempt to make our mathematical
function correspond to reality. Suppose that we define a probability space whose elements are “the coin lands heads up” and “the coin lands tails up.” It is reasonable to define:
• S = {“the coin lands heads up”,“the coin lands tails up”}.
