1. 1.3 Frequency and probability 1.The definition and properties of frequency 2.The definition and properties of probability

2. 1.The definition and properties of frequency 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 Definition: Properties of frequency:

3. Example: A coin is tossed 5times 、 50times 、 500time ， this experiment is repeated 7 times.Observe the number of Head appear and frequency. With the increase of n, the frequency f presents stability

4. Probability 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.

5. 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

6. 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( ) = 1 – P(A). 4.If 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)