Probability Theory-Probability Summary of Events
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Preface
I haven't written a summary of the study of probability theory for many days. Although I have been watching the video study recently, because I have to prepare for a game, a lot of spare time is used to prepare for the game/(ㄒoㄒ)/~~. The preparations for the game are almost done, and it just happened to be fine today, so let’s study the probability theory and sort it out (I feel that I have forgotten some). Today I will sort out the second lecture of probability theory-the probability of events.
1. Elementary description of probability
Definition: In probability theory, the quantitative index used to describe the probability of an event is called the probability of an event. The probability of event A is represented by the notation P(A).
Nature: Since an inevitable event must occur in each test, or that the probability of its occurrence is 100%, its probability is 1. The probability of an impossible event happening is zero. So its probability is 0. That is: 0<=P(A)<=1.
Leading out: Determining the probability of an event is one of the most basic problems in probability theory. Below we will give a method for calculating the probability of an event in two types of specific experiments-classical and geometric probabilities.
2. Classical Probability
1. Classical Probability and Classical Definition of Probability
Two characteristics of classical probabilities:
(i) The sample space of the experiment has only a limited number of sample points.
(Ii) The probability of occurrence of each sample point in the experiment is equal.
Calculation of Classical Probability Type:
Generally, suppose that test E is a classical probability type, its sample space Ω contains n sample points, and the number of favorable sample points for event A is m, then:
P(A)=the number of favorable sample points of A/ The total number of sample points in Ω=m/n
P(A)=the number of basic events included in type A/total number of basic events/total number of basic events
Classical Probability Type Properties:
(1) Non-negativity: 0<=P(A)<=1
(2) Normative: P(Ω)=1, P(fai/empty set=0)
(3) Limited addition Property: A1...An is incompatible with each other: P(A1+A2+A3...An)=P(A1)+...+P(An)
2. Basic calculation principle and permutation and combination formula
(1) Basic technical principle
Permutation and combination: several types of schemes, addition
Principle of multiplication: several steps, multiplication
Arrangement:
1) No repeated arrangement
Take out m different arrangements from n different elements
Pnm=n(n-1)(n -2)…(n-m+1)=n!/(nm)!
Full arrangement: Pnn=n(n-1)x…x3x2x1=n!
0!=1, 1!=1x0!, 0 of 0 Power (meaningless) , the specific meaning is:
But (1) 0! =1, 1! =1x0!. (2) P00=0! , Choose 0 out of 0, and only one possibility is not to choose. (3) Pnm=n!/(nm)!, Pnn=n!/0!=n!, 0! =1
2) Repeat permutation
Take m permutations from n different elements (m can be the same) nxnxnxn=m to the power of n
Combination (just find it out): Take out m different elements from n different elements
Cnm=Pmn /m!=n(n-1)…(n-m+1)/m(m-1)…x2x1=n!/m!(nm)!
Cmn=C(nm)n、Cn0=Cnn=1 .
Three, geometric probabilities
Definition: Each basic event of the experiment can be represented by a point in a geometric area Ω, and all basic events can be represented as all points in Ω (the geometric area Ω is called the area corresponding to the test sample space). The experiment can be attributed to randomly throwing a point M in Ω, and it is equally possible for the point M to fall anywhere in Ω (here the so-called equal may refer to any sub-region in Ω, and the probability that the point M falls within it is equal to the The metric of the area is proportional, and has nothing to do with the shape of the area and its position in Ω). Such a test model is called a geometric model .
Example questions:
example one,
example two,
Fourth, frequency and probability
Definition 1.1 : Suppose A is an event of test E. If event A occurs m times in n repeated tests, then the ratio m/n is called the probability of event A occurring in n repeated tests (frequency), denoted as Wn (A).
**Definition 1.2: ** Suppose A is an event of test E. Under the same conditions, test E is repeated n times. When n is sufficiently large, the frequency Wn(A) of event A is always stable at a certain Swing near a value P, call the stable value P of this frequency the probability of occurrence of event A, denoted as P(A)=P.
Relationship: Wn(A)→P. Wn(A) is the test result and P is the intrinsic property, which exists objectively before the frequency.
5. The axiomatic definition and nature of probability
to sum up
Today, I will put it all together and summarize the probability of the event. I hope I can continue to work hard (ง •_•)ง.