Combinatorics/Probabilities based on math for college(Математика, Богомолов, Самойленко, среднее профессиональное образование)

 

Combinatorics

Set

Collection of elements(in alg for example numbers)

Can be empty, singleton, finite and infinite sets

*inifinite set - for example, type of numbers(real, positive, etc…)

Sequence

Collection of numbers with a specific order(for example progressions)

*infinite set also a sequence(for example arithmetic progression)

Factorial

Product of all positive numbers

n! = n * (n-1) * (n-2) * (n-3) * … | while (n-x)>0

*for example 5! = 5 * 4 * 3 * 2 * 1 = 120

Elements of combinatorics

*possible actions with sets

Variations

From the set with n-length, it is possible to create new sets with rules - they have at least one different element, or order. Formula for calc all variations:

A(n m) = n*(n-1)*(n-2)*...(n-(m-1)) = n! / (n - m)!

Permutations

Special cases of variations when m=n, creates sets with the same length but with different order:

P(n) = A(n m) = n!

Combinations

Action of creating sets with at least one different element, the order do not have sense:

C(n m) = A(n m) / Pm = 1 / (n-m)!

Probabilities

Elements of probabilities

Event

Result of some action. Results can be predicted.

Random event

An event can be successful(true,1)/failure(false,0)/etc…

Certain event

Event will be a successful

Impossible event
Event will be a failure

Incompatible events

For multiple events: if first event is true then second is false, otherwise also

Independent events

For multiple: if first event true then second can be true/false 

Opposite events

For multiple: they describe one event and incompatible

Theorem of the incompatible probabilities sum

Probability of one successful from incompatible events 

P(A+B)=P(A)+P(B)

P(A1+A2+...Ak) = P(A1)+P(A2)+...+P(Ak)

Theorem of the independent probabilities sum

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

*P(AB) probability when they both successful

P(A+B+C)=P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC)

Theorem of the opposite probabilities sum

P(A)+P(B)=1

Theorem of the independent probabilities product

Probability when two successful from both

P(AB)=P(A)*P(B)

P(A1A2…An)=P(A1)*P(A2)*...*P(An)

Theorem of the incompatible probabilities product

Conditional probability

Probability of second(A) after first(B) is successful P(A/B)

P(AB)=P(A)*P(B/A)=P(B)*P(A/B)