1. Теорема добутку ймовірностей та наслідки з неї.

Theorem of product

The probability of the product of two dependent events A and B is equal to the probability of one of them multiplied be conditional probability of the other.

P (AB) = P(A)*P_A(B) – if A appears 1^st

P (AB) = P(B)*P_B(A) – if B appears 1^st

Proof

Let n be the total number of incompatible equally possible outcomes

m – the number of outcomes leading to occurrence of the event A

l – number of outcomes leading to occurrence of the event B

k - the number of outcomes leading to occurrence of the event A and B together.

P(A*B) = k/n=(k*m)/(n*m)=(m/n)*(k/m)=P(A)*P_A(B)

P(A*B)=(k*l)/(n*l)=(l/n)*(k/l)=P(B)*P_B(A)

Consequence

A₁,A₂,A₃…A_n – dependent events, then the probability of their product P(A₁,A₂…A_n) equals P(A₁)*P_A1(A₂)*P_A1A2(A3)*…*PA₁A₂…_An-1(A_n)

Properties of conditional probability:

1. 0<=P_A(B)<=1

2. If A and B are independent , then

3. P(A*B)=P(A)*P(B), P_A(B)=P(B)

2. Теорема суми ймовірностей та наслідки з неї

The addition theorem

The probability of the sum of two events A and B is equal to the sum of their probabilities minus the probability pf their product.

P(A+B)=P(A)+P(B)-P(A*B)

Proof

n – total number of equally probable and incompatible outcomes.

m – number of outcomes leading to the occurrence of event A.

l – number of occurrence of the event B

k – number of outcomes leading to the occurrence A and B together, when the probability of sum A and B is equal by classic definition.

P(A+B) = (m+l-k)/n=m/n +l/n-k/n = P(A)+P(B)-P(A*B)

Consequence 1

If A1,A2,…An from the full group of events, then the probability of their sum is 1.

P(A1+A2+…An)=P(A1)+P(A2)+…+P(An)=1

Consequence 2

If events A and А̅ are opposite, then the probability of their sum is equal:

P(A) +P(A̅)=1

3. Теорема (формула повної імовірності)

Formula of total probability

Let event A can occur only together with events H₁, H₂, H₃,…Hn (with one of this events), H₁, H₂, H₃,… Hn will call hypothesis.

Probabilities of hypothesis are known P(H₁),P(H₂),P(H₃),…P(H_n) and also conditional probabilities of event A is known P_H1(A), P_H2(A),… P_Hn(A).

Events H₁, H₂, H₃,… H_n form full group of probabilities.

P(H₁)+P(H₂)+…+P(H_n)=1

Event A can occur only together with H₁, H₂, … H_{n ,} consequently event A will be the sum of following events.

A=A*H₁+A*H₂+A*H₃+…+A*H_n

To find P(A) we use the theorem of sum of probability. These events are incompatible.

P(A)=P(A*H₁)+P(A*H₂)+…+P(A*H_n)

Let’s use the theorem of product: events A and H1,H2,…Hn are dependent, so

P(A)= P(H₁)*P_H1(A)+ P(H₂)*P_H2(A)+…+ P(H_n)*P_Hn(A) –formula of total probability.

4. Теорема (формули Байєса).

Bayes formula

In practice we are interested in full group of incompatible vents H₁,H₂,…H_n, which probabilities are known P(Hi), i=1̅̅..̅n. These events are not observable, but we may observe the conditional probabilities. If event A already appeared, after event A appeared we can observe conditional probabilities of hypothesis.

P_A(H₁),P_A(H₂),…,P_A(H_n)

P_A(H₁)=P(H₁)*P_H1(A) ; P_A(H₂)= P(H₂)*P_H2(A)

P(A) P(A)

Or using total formula of probability:

P_A(H₁)=P(H₁)*P_H1(A)

∑P(Hi)*P_Hi(A)

^i=1…n