§2 Frequency and probability 2.1The definitions and properties of frequency and properties

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:

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

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.

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( ) = 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)

2.2 Equally Likely Outcomes

The outcomes of a sample space are called equally likely if all of them have the same chance of occurrence. It is very difficult to decide whether or not the outcomes are equally likely. But in this tutorial we shall assume in most of the experiments that the outcomes are equally likely. We shall apply the assumption of equally likely in the following cases ： (1)Throw of a coin or coins: When a coin is tossed, it has two possible outcomes called head and tail. We shall always assume that head and tail are equally likely if not otherwise mentioned. For more than one coin, it will be assumed that on all the coins, head and tail are equally likely (2) Throw of a die or dice: Throw of a single die can be produced six possible outcomes. All the six outcomes are assumed equally likely. For any number of dice, the six faces are assumed equally likely. (3) Playing Cards: There are 52 cards in a deck of ordinary playing cards. All the cards are of the same size and are therefore assumed equally likely.

If an experiment has n simple outcomes, this method would assign a probability of 1/n to each outcome. In other words, each outcome is assumed to have an equal probability of occurrence. This method is also called the axiomatic approach. Example 1: Roll of a Die S = {1, 2, · · ·, 6} Probabilities: Each simple event has a 1/6 chance of occurring. Example 2: Two Rolls of a Die S = {(1, 1), (1, 2), · · ·, (6, 6)} Assumption: The two rolls are “independent.” Probabilities: Each simple event has a (1/6) · (1/6) =1/36 chance of occurring.

Theorem1.1 In classical probability counting is used for calculating probabilities. For the probability of an event A we need to know the number of outcomes in A, k, and if the sample space consists of a finite number of equally likely outcomes, also the total number of outcomes, n.

E1: A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color?

E2: A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number? Roll of a Die P(even) = 3/6 P(low) = 3/6 P(even and low) = P({2}) = 1/6 P(even or low) = 3/6 + 3/6 − 1/6 = 5/6 P({1} or {6}) = 1/6 + 1/6 − 0 = 2/6

E3: A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?