1 3 Frequency and probability 1 The definition

1. 3 Frequency and probability 1. The definition and properties of frequency 2. The definition and properties of probability

1. The definition and properties of frequency Definition: Consider performing our experiment a large number , n times and counting the number of those times when A occurs. The frequency of A is then defined to be Properties of frequency:

Example: A coin is tossed 5 times、500 time，this experiment is repeated 7 times. Observe the number of Head appear and frequency. With the increase of n, the frequency f presents stability

Probability of an event A of a repeatable experiments is given by Experiment ‘tossing a coin’: The relative frequency of the event ‘head up’ as the function of the number of trials.

Probability Axioms Definition 1. 9 A Probability measure on a sample space S is a function P which assigns a number P(A) to every event A in S in such a way that the following three axioms are satisfied: Axiom 1. P(A) ≥ 0 for every event A. Axiom 2. P(S)=1. Axiom 3. Countable additivity可列可加性. i. e. , if A 1, A 2, …… is an infinite sequence 无穷序列 of mutually exclusive (disjoint) event两两互不相容事件 then

Properties of Probability 1. P(Ø) = 0, P(S) = 1. 2. If A and B are disjoint events then (disjoint or mutually exclusive means A∩B = Ø) 3. For any event A, P( 4. If ) = 1 – P(A). then P(A - B) = P(A) - P(B) and 5. For any A and B, P(A∪B ) = P(A) + P(B) − P(A∩B ) P(A∪B ) P(A) + P(B)