Introduction

1) Calcolo of the probabilities :

Discipline that concurs to analyze to the phenomena and the aleatory amounts, constructing of a model.

 

2) Together :

Collection of objects, realizes or separated, sayings elements.

 

3) Partition of together :

Draft of a class of sottosets not empty that covers the entire entirety without superimpositions.

 

4) fundamental Principle of the combinatorio calculation :

If a procedure can be realized in n₁ various ways, and if, after this procedure, a second procedure can be realized in n₂ various ways, and if, after this second procedure a third procedure can be realized in n₃ various ways, and therefore via ; then the number of ways in which the procedure it can be realized in the indicated order is

n₁ * n₂ * n₃ *.... * n_{n .}

 

5) Dispositions :

Groups objects order obtained to you taking in a data order m on N. It can be calculated directly using the fundamental principle of the combinatorio calculation equivalent to

 

6) Permutations :

Groups the number of the permutations orders obtained to you taking in a data N order objects on N. is n !.

 

7) Permutations with repetitions :

There are of the permutations in which some objects are equal between they and therefore it does not give place to distinguished permutations, in this case is necessary to divide for the number of dispositions that everyone of these objects invalid.

 

8) Write the value of the binomial coefficient :

Ha where the numerator is obtained immediately multiplying for the natural number more small stopping itself to the number given from the difference between 9 and 4 increased of 1.

 

9) Enounce a theorem much regarding profit the binomial coefficient :

on condition that to = b c

 

10) Combinations :

Groups do not order obtained to you taking m objects on N, their number are equal to the binomial coefficient

 

11) accidental Experiment :

Procedure of observation of the final state on the system subordinate to the experiment, than supposes ripetibile infinitely a number of times with the same modalities of execution.

 

12) With universal or Sample space :

With of all the possible ones he turns out you of an accidental experiment.

 

13) Event :

It is with of turns out to you.

 

14) incompatible Events :

Two events are incompatible if their intersection is an impossible event, that is the events do not have turn out to you in common.

 

15) fundamental Axioms of Kolmogorov :

to)    P(A) it is a positive or null number.

b)    the sure event has unitary probability.

c)    if 2 events are incompatible, the probability of the event union is equal to the sum of the probabilities of the single events P(A?B) = P(A) ? P(B)

 

16) Relative frequency :

Draft of the relationship between the number of times n(A) in which an element has itself like result of with To and the number n of tests of esperimento the .

 

17) classic Definition of probability :

The probability of an event To is the relationship between the possible ones turns out to you favorable to the event To n(A) and the number of the possible ones turns out to you n

18) If 0 are with the empty one ž P(0) = 0 :

19) conditioned Probability :

If To and B they are 2 events of a sample space S with P(B) ¹ 0, conditioned probability of To regarding B is defined, and it is indicated with P(A|B), the relationship meaning with ci² that the probability that the event To is taken place, once that the B event has been taken place is given from the relationship of the probability of the intersection and the probability of the B event

 

20) Property of the conditioned probability :

to)    P(A|B) is a positive number

b)    P(S|B) = 1

c)    If To and B are events incompatibili then P(A B | M) = P(A|M) P(B|M)

 

21) statistically independent Events  :

Two events say statistically independent if verification eguaglianza P(AB) = P(A) * P(B).

 

22) Property of the independent events :

to)    P(A B) = P(A) P(B) - P(A)*P(B)

b)    also To and B they are independent

c)    if To, B, C are independent events, also To and BC it they are

d)    if To, B, C are independent events, also To and B C it they are

 

23) Theorem of the probability total :

The probability of a B event defined on a sample space S can be expressed in term of conditioned probabilities considering one partition of S.

P(B) = P(A₁)*P(B|To₁) ....... P(A_m)*p(b|To_m)

 

24) Theorem of Bayes :

It is a useful theorem in all those cases in which there is a partizionato sample space and to every partition a probability is associated and the probability is wanted as an example to be known that the piece produced from the B machine has also the typical characteristic of To.

 

25) Bernoulliane Tests :

Draft of with of tests, between independent they, in which there are 2 single ones turns out possible to you.

 

26) Probabilities of having K succeeded in a data order :

p^kq^{n - k}

 

27) succeeded Probabilities of having K in a whichever order :

 

28) rare Event :

An event says rare if verification with one probability much minor of 1.

 

29) Theorem of Poisson :

It concurs to us to quantify the probability easy that one takes place k times a rare event To, in fact being n the number of the tests and p the probability of the rare event To. In such a way it is simplified I use it of the formula of Bernoulli.