انت هنا الان : شبكة جامعة بابل > موقع الكلية > نظام التعليم الالكتروني > مشاهدة المحاضرة

Probability Theory (Part 3)

Probabilistic Experiment

A Probabilistic Experiment is a situation in

Which More than one thing can happen

 The outcome is potentially uncertain

The Sample Space

The Sample Space of a probabilistic

experiment E is the set of all possible experiment E is the set of all possible

outcomes of E.

Examples:

E1 = Toss a coin, observe whether it is a

Head

(H) or a Tail (T)

?1 = {H, T}Examples:

E2 = Toss a fair die, observe the outcome.

?2 = {1, 2, 3, 4, 5, 6}

E3 = Toss a fair coin 5 times, observe the

number of heads.

?3 = ? (C.P.)

Examples:

E4 = Toss a fair coin 5 times, observe the sequence

of heads and tails. of heads and tails.

?4 ={HHHHH, HHHHT, HHHTH, HHHTT,

HHTHH, HHTHT, HHTTH, HHTTT, ….

Even with very simple situations, the Sample Space

can be quite large. Note that more than one

Probabilistic Experiment may be defined on the

same physical process.Elementary Events vs. Compound Events

The Elementary Events in a Sample Space are the finest

possible partition of the sample space.

Compound Events are the union of elementary events.

Example:

Toss a fair die. (E2)

The elementary events are 1,2,3,4,5 and 6. The elementary events are 1,2,3,4,5 and 6.

The events “Even” = {2,4,6}, “Odd” =

{1,3,5} are examples of compound events.The Axioms of Relative Frequency

EventRelative

Freq

6 1/6

5 1/6

41/6

3 1/6 3 1/6

21/6

11/6

(>3) 3/6

Even 3/6

odd3/6

Even U4/6

Even U Odd1Fundamental Theorems of Probability

Theorem 1.

Proof. For all events A, .

Pr (?) = 0

A? ? = ?

So the 3rd axiom applies, and we have

Pr(A??) = Pr(A) + Pr(?) = Pr(A)

But, for any set A, , so by

subtraction, we have the result.

Theorem 2.

?

Theorem 2.

Proof. For all events A, .

But, for any set A, A ? A = ? , so

Pr(A) = 1? Pr(A)

A? A =?, so Pr(A? A) = Pr(A) + Pr(A)

Pr(?) = 1 = Pr(A) + Pr(A)

The result then follows by

subtraction.

Theorem 3.

Proof. Someone from the class will prove this well known result.Thank you!