1. Randomized trials

Trials with the following characteristics are called randomized trials

  1. It can be repeated under the same conditions
  2. There is more than one possible outcome for each test, and all possible outcomes for a definite test can be achieved;
  3. It is not certain which result will occur until a trial is carried out

2. Sample space

  • The set of all possible outcomes of a randomized trial EEE is called the sample space of the EEE, and is denoted as SSS

  • The elements of the sample space, each result of the EEE, are called sample points

3. Random events

  • The subset of SSS of sample space of test EEE is EEE random event, or event for short
  • This event is said to occur if and only if a sample point in this subset is present in each trial
  • A single set of sample points is called a basic event
  • The sample space SSS contains all sample points and is called necessary events
  • The empty set ∅\empty∅, which does not contain any sample points, does not occur in each experiment and is called an impossible event

4. The relation between events and the operation of events

4.1 Relationship between Events

  1. The sub-event: A⊂BA \subset BA⊂B, means that AAA occurs, then BBB must occur
  2. Equal events: A⊂BA \subset BA⊂B and B⊂AB \subset AB⊂A => A=BA=BA=B
  3. And events: A ∪ B = {x ∣ x ∈ Aorx ∈ B} \ cup B = \ \ {x | x in A quad or \ quad, x \ B \} in A ∪ B = {x ∣ x ∈ Aorx ∈ B} AAA and BBB at least one
  4. Product events: A studying B = {x ∣ x ∈ Aandx ∈ B} \ cap B = \ {x | x \ \ A in quad and \ quad, x \ B \} in A studying B = {x ∣ x ∈ Aandx ∈ B} AAA and BBB occurred at the same time
  5. Bad events: A – B = {x ∣ x ∈ Aandx ∉ B} A – B = \ {x | x \ \ A in quad and \ \ quad x not \ B \} in A – B = {x ∣ x ∈ Aandx  ∈ B} AAA, BBB will not occur
  6. Exclusive events: A∩B=∅A \cap B= \emptysetA∩B=∅
  7. Inverse event/opposite event: A∪B=SandA∩B=∅A \cup B=S \quad and \quad A \cap B= \emptysetA∪B=SandA∩B=∅ AAA and BBB have only one ∪B= (4) which is ‾=S−B \overline {A} = S – \ overline {B} A = S – B

4.2 Event Operation

  1. Union B=B union AA \cup B=B union AA =B union A; A∩B=B∩AA \cap B=B \cap AA∩B=B∩A

  2. Associative law (A ∪ B) ∪ C = A ∪ ∪ C (B) (A \ cup B), C = A cup, cup (B \ C cup) (∪ B) ∪ C = A ∪ ∪ C (B); (A studying B) C = A studying studying studying C (B) (A \ cap B) \ cap C = A \ cap (B \ C) cap (A studying B) studying C = A studying studying C (B)

  3. Distributive law (A ∪ B) studying C = (A studying C) ∪ studying C (B) (A, cup B), C = (A \ cap C) of the cap \ cup \ cap (B), C (A ∪ B) studying C = (A studying C) ∪ studying C (B); (A studying B) ∪ C = (A ∪ C) studying ∪ C (B) (A \ cap B) \ CPU C = (A, cup C), cap (B \ C) cup (A studying B) ∪ C = (A ∪ C) studying ∪ C (B)

  4. A∪B= an ‾=A ∪B ∩B \overline{A \cup B}=\overline{A} \cap \overline{B}A∪B=A∩B;

    A studying ‾ B = A ‾ ∪ ‾ \ overline B = {\ cap B} \ overline {A} \ cup \ overline {B} A studying B = ∪ B

5. Frequency and probability

5.1 the frequency

  • Under the same conditions, NNN tests are carried out. In these NNN tests, the frequency of event AAA is called nAn_AnA, and the ratio nA/nn_A/nnA/n is called the frequency of event AAA, denoted as Fn (A) F_N (A) Fn (A).

5.2 probability

  • Let EEE be A random trial, and SSS be its sample space. For each event AAA of EEE, A real number is assigned, denoted as P(A)P(A)P(A) P(A)P(A), which is called the probability of event AAA. Probability functions have the following properties

    1. Nonnegative: For each event AAA, there is P(A)>=0P(A)>=0P(A)>=0

    2. Normative: For inevitable events, P(S)=1P(S)=1P(S)=1

    3. List additivity: Let A1,A2,A3… A_1,A_2,A_3,\cdotsA1,A2,A3… be two mutually incompatible events. Namely AiAj = ∅, I indicates j, I, j = 1, 2, 3,.. A_iA_j = \ emptyset, I \ neq j, \ quad I, j = 1, 2, 3, \ cdotsAiAj = ∅, I  = j, I, j = 1, 2, 3,…


      P ( A 1 ∪ A 2 ∪ A 3 …   ) = P ( A 1 ) + P ( A 2 ) + P ( A 3 ) + … P(A_1 \cup A_2 \cup A_3 \cdots )=P(A_1)+P(A_2)+P(A_3)+\cdots

  • The nature of the


    1. P ( ∅ ) = 0 P(\emptyset)=0
    2. Limited additivity If A1, A2, A3,…, AnA_1, A_2, A_3, \ \ cdots, A_nA1, A2, A3,…, the An is two incompatible event, has P (A1 ∪ A2 ∪ A3.. ∪ An) = P (A1) + P (A2) + P (A3) +. + P (An) P (A_1 \ CPU A_2 \ CPU A_3 \ cdots \ CPU A_n) = P (A_1) + P (A_2) + P (A_3) + \ cdots + P (A_n) P (A1 ∪ A2 ∪ A3.. ∪ An) = P (A1) + P (A2) + P (A3) +. + P (An)
    3. Let A,BA,BA, and B be two events, and as A⊂BA \subset BA⊂B, P(B−A)=P(B)−P(A); P P P (B) (A) P (B – A) = P (B) – P (A); \ \ quad P (B) geq P (A) P (B – A) = P (B) – P (A); P P P (B) (A)
    4. For any event AAA, P(A)≤1P(A) \leq 1P(A)≤1
    5. The probability of adverse events For any event AAA, P (A ‾) = 1 – P (A) P (\ overline {A}) = 1 – P (A) P (A) = 1 – P (A)
    6. Addition formula For any two events A, BA, BA, B is P (A ∪ B) = P (A) + P (B) – P (AB) P (A \ cup B) = P (A) + P (B) – P (AB) P (A ∪ B) = P (A) + P (B) – P (AB) 1 o. 1 ^ o. 1 o. This property can be extended to several events, such as A1,A2,A3A_1,A_2,A_3A1,A2,A3 are any three events, Is P (A1 ∪ A2 ∪ A3) = P (A1) + P (A2) + P (A3) – P (A1A2) – P (A1A3) – P (A2A3) + P (A1A2A3) P (A_1 \ CPU A_2 \ CPU A_3) = P (A_1) + P (A_2) + P (A_3) – P (A_1A_2) – P (A_1A_3) – P (A_2A_3) + P (A_1A_2A_3) P (A1 ∪ A2 ∪ A3) = P (A1) + P (A2) + P (A3) – P (A1A2) – P (A1A3) – P (A2A3 )+P(A1A2A3) 2o.2^o.2o. In general, for any NNN events, A1,A2,A3,… AnA_1,A_2,A_3,\cdots,A_nA1,A2,A3,… An, Can use the induction card P (A1 ∪ A2.. ∪ An) nP = ∑ I = 1 (Ai) – + ∑ ∑ 1 I < j nAiAj or less or less acuities were I < j < 1 k or less nP (AiAjAk) +. + (1) – n – 1 P (A1A2… An) P (A_1 \ CPU A_2 \ cdots \ CPU A_n)=\sum_{i=1}^{n}{P(A_i)} – \sum_{1 \le i

6. Classical probability model

  • 1) The sample space of the test contains only a limited number of elements. 2) The probability of each basic event in the test is the same, and the test with these two characteristics is called equal probability probability type, which is the main research object in the early development of probability theory, so it is also called classical probability type


    P ( A ) = ∑ j = 1 k P ( { e i j } ) = k n = A The number of base events that are included S The total number of base events in (A) = P \ sum_ {j = 1} ^ {k} {P (\ {{e_i} _j \})} = \ frac {k} {n} = \ frac {contains A number of basic event} the total number of basic events in {S}

Conditional probability

  • For the general classical scheme, let the total number of basic events in the test be NNN, the number of basic events contained in AAA is m(m>0) M (m>0)m(m>0)m(m>0), the number of basic events contained in ABABAB is KKK, Have P (B ∣ A) = k = k/nm/n = P (AB) P (A) P (B | A) = \ frac {k} {m} = \ frac {k/n} {m/n} = \ frac {P (AB)} {P} (A) P (B ∣ A) = mk = m/nk/n = P (A) P (AB).

  • Define let A,BA,BA,B be two events, > 0, P (A) P (A) > 0, P (A) > 0, said P (B ∣ A) = P (AB) P (A) P (B | A) = \ frac {P (AB)} {P} (A) P (B ∣ A) = P (A) P (AB) AAA for the event under the condition of incident BBB of conditional probability, has the following properties:

    1. The negative: BBB for each event, with P (B ∣ A) acuity 0 P \ ge 0 P (B | A) P (B ∣ A) 0
    2. Normative: for A certain event SSS, P = 1 (S ∣ A) P (S | A) = 1 P (S ∣ A) = 1
    3. Column additivity: set
      B 1 . B 2 . … B_1,B_2,\cdots       Is twoincompatibleOf events, there are
      P ( ⋃ i = 1 up ∣ A ) = ∑ i = 1 up P ( B i ∣ A ) P(\bigcup_{i=1}^{\infty} |A)=\sum_{i=1}^{\infty}{P(B_i|A)}

      For any event B1,B2B_1,B_2B1,B2, P (B1 ∪ B2 ∣ A) = P (B1 ∣ A) + P (B2 ∣ A) – P (B1B2 ∣ A) P (B_1 \ CPU B_2 | A) = P (B_1 | A) + P (B_2 | A) – P (B_1B_2 | A) P (B1 ∪ B2 ∣ A) = P (B1 ∣ A) + P (B2 ∣ A) – P (B1B2 ∣ A).

  • The multiplication theorem

    By the definition of conditional probability, P (A) > 0, P (A) > 0, P (A) > 0, has P (AB) = P (B ∣ A) P (A) P (AB) = P (B | A) P (A) P (AB) = P (B ∣ A) P (A), the type is called the multiplication formula

    • Generally, A1, A2, A3,…, AnA_1, A_2, A_3, \ \ cdots, A_nA1, A2, A3,…, An n events, n p 2 n \ ge 2 n 2 or more, And P (A1A2A3.. An – 1) > 0, P (A_1A_2A_3 \ cdots A_ {}, n – 1) > 0, P (A1A2A3.. The An – 1) > 0, Have P (A1A2A3.. An) = P (An ∣ A1A2A3.. The An – 1) P (An – 1 ∣ A1A2A3.. The An – 2). P (A2 ∣ A1) P (A1) (1) P (A_1A_2A_3 \ cdots A_ {n}) = P (A_n | A_1A_2A_3 \ cdots A_{n-1})P(A_{n-1}|A_1A_2A_3 \cdots A_{n-2}) \cdots P(A_2|A_1)P(A_1) \quad (1) P (A1A2A3.. An) = P (An ∣ A1A2A3.. The An – 1) P (An – 1 ∣ A1A2A3.. The An – 2). P (A2 ∣ A1) P (A1) (1)

      When understanding Equation (1), you can try to analyze it from the right side of the equation on the right side of the equal sign. Such as P (A2 ∣ A1) P (A1) = P (A1A2) P (A_2 | A_1) P (A_1) = P (A_1A_2) P (A2 ∣ A1) P (A1) = P (A1A2), (1) the reciprocal of the third for P (A3 ∣ A1A2) P (A_3 | A_1A2) P (A3 ∣ A1A2), then P (A3 ∣ A1A2) P (A2 ∣ A1) P (A1) = P (A3 ∣ A1A2) P (A1A2) = P (A1A2A3) P (A_3 | A_1A2) P (A_2 | A_1) P (A_1) = P (A_3 | A_1A2) P = (A_1A_2) P(A_1A_2A_3)P(A3)P(A2)P(A1)=P(A3)P(A1A2)=P(A1A2A3), and so on, we can sum up the formula (1)

  • Total probability formula

    • Set SSS as the sample space of EEE, B1,B2,… BnB_1,B_2,\cdots B_nB1,B2,… Bn as a group of events of EEE, if


      1. B i B j = ∅ . i indicates j . i . j = 1 . 2 . …   . n ; B_iB_j = \ empty, I \ neq j, I, j = 1, 2, \ \ cdots, n.
      2. B1 ∪ B2 ∪.. ∪ Bn = SB_1 \ CPU B_2 \ cup \ cdots \ CPU B_n = SB1 ∪ B2 ∪. ∪ Bn = S,

      BnB_1,B_2,\cdots B_nB1,B2,.. Bn is a partition of sample space SSS (complete event group).

    • Theorem If the sample space of EEE is SSS and AAA is EEE, B1,B2,… BnB_1,B_2,\cdots B_nB1,B2,… Bn is a partition of SSS, And P (Bi) > 0, (I = 1, 2,…, n), P (B_i) > 0, (I = 1, 2, \ \ cdots, n), P (Bi) > 0, (I = 1, 2,…, n), the P (A) = P (A, B1) ∣ P (B1) + P (b ∣ B2) P (B2), +. + P (A ∣ Bn) P (Bn) P (A) = P (A | B_1) P (B_1) + P (A | B_2) P (B_2) + \ cdots + P (A | B_n) P (B_n) P (A) = P (A ∣ B1) P + P (B1) A given B2 P of B2 plus — plus P of A given Bn P of Bn is called the total probability formula

    • Theorem If the sample space of EEE is SSS and AAA is EEE, B1,B2,… BnB_1,B_2,\cdots B_nB1,B2,… Bn is a partition of SSS, And P (A) > 0, P (B) > 0, (I = 1, 2,…, n), P (A) > 0, P (B) > 0 \ quad (I = 1, 2, \ \ cdots, n), P (A) > 0, P (B) > 0, (I = 1, 2,…, n), then

      P (Bi ∣ A) = P (A ∣ Bi) P (Bi) ∑ j = 1 np (A Bj) ∣ P (Bj) I = 1, 2,…, n. (2) P (B_i | A) = \ frac {P (A | B_i) P (B_i)} {\ sum \ limits_ ^ {j = 1} {n} {P (A | B_j) P (B_j)}} \ quad I = 1, 2, \ \ cdots, n. \ quad (2) P (Bi ∣ A) = j = 1 ∑ nP (A Bj) ∣ P (Bj) P (A ∣ Bi) P (Bi) I = 1, 2,…, n. (2) the type called bayesian formula (Bayes theorem)

    • The probability obtained from previous experience and analysis is called prior probability, such as P(Bi)P(B_i)P(Bi) in Equation (2).

    • The posterior probability of a random event or an uncertain event is the conditional probability obtained after considering and giving relevant evidence or data

    Reference formula 2 can be simply understood as

    AAA: Results of events (observational data)

    BBB: Influencing factors of the event

    P (B ∣ A) P (B | A) P (B ∣ A) : A posteriori probability

    P(B)P(B)P(B) : prior probability

Independence 8.

  • Set A, BA, BA, B two events, if meet the equation P (AB) = P (A) P (B) P (AB) = P (A) P (B) P (AB) = P (A) P (B), called event A, BA, BA, B are independent of each other, hereinafter referred to as A, BA, BA, B independently

  • Theorem 1 set A, BA, BA, B two events, and P (A) > 0, P (A) > 0, P (A) > 0, if A, BA, BA, B are independent of each other, then P (B ∣ A) = P (B) P (B | A) = P (B) P (B ∣ A) = P (B) and vice versa.

  • Theorem 2 If AAA and BBB are independent of each other, then the following team events are also independent of each other:

    • (4) AAA and B‾\overline BB, BBB is with A‾\overline AA, A (4) overline A and overline BA and B

  • Let’s say that A,B,CA,B,CA,B, and C satisfy this equation


    P ( A B ) = P ( A ) P ( B ) . P ( B C ) = P ( B ) P ( C ) . P ( A C ) = P ( A ) P ( C ) . P ( A B C ) = P ( A ) P ( B ) P ( C ) } \ left \ begin {array} {LCL} P (AB) = P (A) P (B), \ \ P (BC) = P (B) (C), P \ \ P (AC) = P (A) P (C), \\ P(ABC)=P(A)P(B)P(C) \\ \end{array} \right\}

    Then events A, B, CA, B, CA, B, C are said to be independent of each other. This definition has the following corollary

    • If the event A1, A2,…, An n p (2) the A_1 and A_2, \ \ cdots, A_n (n \ ge2) A1, A2,…, An independent (n 2 or higher), is one of k (k, n) or less or less 2 k \ \ le le k (2 n) k (2 k, n) or less or less events are independent of each other
    • If NNN event A1, A2,…, An (n 2) or more A_1 and A_2, \ \ cdots, A_n (n \ ge2) A1, A2,…, An independent (n 2) or more, Then change any number of events in A1,A2,…,An(n≥2)A_1,A_2,\cdots,A_n(n\ge2)A1,A2,…,An(n≥2) into their opposite events, and the obtained NNN events are still independent of each other

  • If NNN event A1, A2,…, An (n 2) or more A_1 and A_2, \ \ cdots, A_n (n \ ge2) A1, A2,…, An independent (n 2) or more, with P (nai) ⋂ I = 1 = ∏ I = 1 np (Ai); P (nai) ⋃ I = 1 = 1 – ∏ I = 1 np ‾ (Ai) = 1 – ∏ I = 1 n (1 – P (Ai)) P (\ bigcap_ {I = 1} ^ {n} {A_i}) = \ prod_ {I = 1} ^ {n} {P (A_i}); \quad P(\bigcup_{i=1}^{n}{A_i}) =1- \prod_{i=1}^{n}{P(\overline{A_i})}=1- ^ \ prod_ {I = 1} {n} {} (1 – P (A_i)) P (nai) ⋂ I = 1 = ∏ I = 1 np (Ai); P (nai) ⋃ I = 1 = 1 – ∏ np (Ai) I = 1 = 1 – ∏ I = 1 n (1 – P (Ai))