Lektsii (1) / Lecture 7

ICEF, 2012/2013 STATISTICS 1 year LECTURES

LECTURE 7

October 13, 2012

CHAPTER 3

PROBABILITY

EXPERIMENT WITH UNCERTAIN OUTCOME

1.The result of an action cannot be predicted for sure.

2.An experiment can be repeated (at least potentially) many times.

Examples

1.Tossing a coin: the results are head or tail but cannot be predicted.

2.Rolling a die. There are six possible numbers on a face side, and again the result is not predictable.

3.Every day you use some bus that has 10 min intervals. Your waiting time (in min) may be any number between 0 and 10.

4.Making a budget research you randomly select a family from a population. The monthly income is an example of uncertain outcome.

An event that may occur or may not occur in an experiment with uncertain outcome is called a random event.

How to define the probability of a random event?

Consider the simplest example when the experiment is the tossing a fair die. The outcome is the number of point on the face side. The event is “six points on the face side”. If we throw a die many times, say 600 times, and then we get the frequency of the event under consideration. For example this value may be 116, or 95, or even 120. In fact, the number of outcomes with six on the face side is random and may be different. The relative frequency (frequency/600) is random value as well. But if the number of trials infinitely increases the relative frequency tends to 1/6. This number is called the probability of the event “six points on the face side” when you toss a fair die one time.

Generally,

The probability of an event is the limiting relative frequency

MODELS

In practice we usually cannon repeat an experiment infinitely many times in order to find the probability of some event. The most appropriate outlet is to construct an adequate mathematical model. And the probability deals just with the mathematical models of the experiments with uncertain outcomes.

1. The space of elementary outcomes

Go back to the tossing a die. Consider two events: A = {the number of points is 4}, B = {the number of points is an even (чётное) number}. The event B occurs when the outcomes are 2, or 4, or 6, while the event A cannot be splitted into “more elementary” outcomes.

So, one of the main concept of the mathematical model of an experiment with uncertain outcomes is so called the space of elementary outcomes or sample space. Each point of this

space is some outcome of the experiment that cannon be divide into “more elementary” outcomes.

Examples

1.Rolling a die one time: the space of elementary outcomes consists of six points: {1, 2, 3, 4, 5, 6}.

2.Rolling two dice, black die and red die: the elementary outcome may be represented as a pair of numbers (a,b) , where a is the number of points on the black die, and b is the number of

points on the red die. An a may be any integer from 1 to 6, and similarly for b. So, the total number of outcomes is 36.

3. Rolling a die until the first appearance of “6”. Let’s denote the appearance of “6” by 1 and any other outcome by 0. Then the elementary outcomes may be described as follows:

s	= (1), s	2	= (0,1), s = (0,0,1),..., s	n	= (0,0,...,0,1),... . In this example the sample space is an
1			3

n−1

infinite set.

4. Waiting for a bus. The elementary outcome is any number from 0 to 10. The space of elementary outcomes is the segment [0, 10].

Let S be some outcome space. What is a random event?

An event A is a subset of the outcome set S.

2. Probability of events

Let S ={s1, s2 ,..., sn ,...} be some outcome space. Let’s assume that we assign some nonnegative number pi = Pr(si ), i =1,2,... to each elementary outcome si , such that the following conditions hold:

1) pi ≥ 0 ;	2) ∑pi =1 .
	i

Definition. The number pi is called the probability of the elementary outcome si . Any set of numbers p1, p2 ,..., satisfying 1), 2) is called the distribution on S.

Definition. Let A be some event, i.e. A S . The probability of an event A, is the some of probabilities of elementary outcomes that are contained in A:

Pr(A) = ∑ pi = ∑Pr(si ) .

i:si A si A

Example. The event A ={the number of points on the face side of a die is even} consists of three

elementary outcomes: A ={2, 4, 6}. If a die is fair then Pr(A) =3 1				= 1 .
	6			2
	Compound events
Let A, B be some events. We shall consider the following events:
•	A B − the union of A and B, A or B. The set A B consists of the elements that are
	contained in A or B. This event occurs if A or B occurs.
•	A ∩B − the intersection of A and B, A and B. The set A ∩B consists of the elements that
	are contained in A and B. This event occurs if A and B occurs.
•	A \ B − the complement of B to A, A minus B. The set A \ B consists of the elements that
	are contained in A but do not belong to B. The event A \ B occurs if A occurs and B does
	not occur. The complement S \ A is denoted as		.
		A

Example. Suppose that a student randomly selected from the ICEF students. Let A = {a student is a male}, B = {a student is honours student}. Then A B means that selected student is either

a male or honours student or both. The event A ∩B means that a male honours student was selected, while A \ B means that selected student is a male student but not honours student.

Definition. If A ∩B = then the events A, B are called disjoint.

Properties of probability of events

1.0 ≤ Pr(A) ≤1, Pr(S) =1.

2.If A B then Pr(A) ≤ Pr(B) .

3.Additivity: if A ∩B = then Pr(A B) = Pr(A) +Pr(B) .

4.Pr(A) =1−Pr(A) .

5.If A B then Pr(A \ B) = Pr(A) −Pr(B) .

6.Generalization of 3:

Pr(A B) = Pr(A) +Pr(B) −Pr(A ∩B) .