Probability statistics: The first chapter of the basic concept of probability theory

Chapter One basic concepts of probability theory

Content Summary:

A. Principle and arrangement of addition and multiplication, and review of Combination

1. The addition principle is set to complete one thing there is a class method (either of these methods can be achieved

The purpose of accomplishing this), if the 1th kind of method, the 2nd class method has the kind, the first kind of method has the kind, then completes this matter to have a total of + + method.

2. Multiplication principle set to complete one thing there is a step (complete each step in sequence to achieve

The purpose of accomplishing this), if the 1th step has a method, the 2nd step has a method, the first step has the method, then completes this matter to have the common method.

3. Arranged

(1) Select and arrange all the elements from a different element in order to form a column, called a permutation of elements from the elements, all the permutations of the elements from the elements are recorded as

；

The arrangement of taking out all the different elements is called the whole arrangement, and the number of permutations is recorded as

=

Provisions.

(2) repeatable arrangement from a different element can be repeated (there is put back) to take a meta

The vegetarian is arranged in a single column, in the order of the number of rows.

(3) The whole arrangement of the different elements has the same, and there is the same, and there is the same, and, the whole arrangement of such elements is called the whole arrangement of the different elements, and its arrangement of the number of species.

(4) The ring arrangement from the different elements of any element is not divided into the first and the same ring arrangement, the number of species.

4. Combination

(1) The general meaning of the combination of each element from a different element in order to take a set of elements, called from the elements of a combination of elements taken out from the elements of all combinations of elements to be counted as

Or

The combination has the following properties:,.

(2) repeatable arrangement from a different element can be repeated (there is put back) to take a meta

The elements are sorted in a group, called a repeatable combination of an element from an element, with all the repeatable combinations of elements taken from each element.

two. Randomized trials and random events

1. Randomised trials (recorded as) if the test (observation or experimental process) satisfies the condition:

(1) can be repeated under the same conditions,

(2) The result of the test is clearly known, and there are many possibilities,

(3) It is not possible to determine which result will appear before each test.

The test is called a randomized trial .

2. Sample space and sample point test all possible results consist of a set of sample spaces , recorded as; each of the possible results of the experiment is called a sample point .

3. A result of random event randomized trials, that is, any subset of the sample space, called a random event , expressed in uppercase letters. It can also be subdivided into

(1) Each non-decomposed result of the basic event randomized trial (single sample point group

Single-point set),

(2) Events consisting of several basic events of complex events (set by several sample points

Combined),

(3) The inevitable event sample space contains all the sample points, it is a subset of itself, in each experiment it always happens, called the inevitable event, still remember,

(4) The Impossible event empty set does not contain any sample points, which as a subset of the sample space, do not occur in each experiment, called The Impossible event, recorded as.

4. The relationship between events and their operations

(1) contains: If an event occurs, it is said to be contained in, or contained, recorded as,

(2) Equality: if and, then the Equals, is recorded as,

(3) and events: events and/or ∪ represent at least one occurrence of events and events that are promoted as follows:

∪∪ ... ∪ indicates that at least one of the events,,..., occurred,

∪∪ ... ∪∪ ... Represents an event,,...,,... At least one of them occurred,

(4) Product event: the product (or intersection) of the event ∩ indicates that events and events occur simultaneously, and is promoted as follows:

∩∩ ... ∩ indicates that an event,,..., occur at the same time,

∩∩ ... ∩∩ ... Represents an event,,...,,... happen at the same time,

(5) Difference event: The event occurs and the event does not occur, the difference is said,

(6) Mutually exclusive events (incompatible): If events and events cannot occur simultaneously, that is, ∩=, it is called a mutex event,

Note: The basic event must be 22 mutually exclusive.

(7) Opposing events (inverse events): In each experiment, "events that do not occur" is called an opposing event of the event, as recorded.

Note: ∪=,=,= and there is a definition, the opposite event must be mutually exclusive event, but the mutex event is not necessarily the opposite event,

(8) The Operation rule of the event

(Ⅰ) Exchange law: ∪=∪,=

(Ⅱ) Binding law: ∪ (∪c) = (∪) ∪,∩ (∩c) = (∩) ∩

(Ⅲ) Distribution law: ∪ (∩c) = (∪) ∩ (∪), ∩ (∪c) = (∩) ∪ (∩)

(Ⅳ) De Morgan Law: =∩,=.

three. The definition and nature of probability

1. The axiomatic definition of probability

The sample space for a randomized trial is, for each event, the definition of a real number

corresponding to it, if the function satisfies the condition

(Ⅰ) nonnegative for each event, there are,

(Ⅱ) Normative,

(Ⅲ) can be added for any 22 mutually exclusive event,,...,,..., there are ∪∪ ... ∪∪ ... =

is called the probability of an event.

2. The nature of probability

(1),

Note: Its inverse is not true, that is, a probability of 0 event is not necessarily an impossible event.

(2) Limited additive to any 22 mutually exclusive event,,...,, there are ∪∪ ... ∪) =,

(3) If, then there is,,

Note: When the condition is not met, the general, but has.

(4) For any event,

(5) For any event,

(6) Addition formula for arbitrary events and, there are

, the promotion is as follows:

∪∪ ... ∪) =+

+