Total probability formula (Bayesian formula)

An event A is caused by three factors: B, C, D, C and D. Find the probability of this event.

p(A)= p(A/B)p(B)+p(A/C)p(C)+p(A/D)p(D)

Where p(A/B) is called conditional probability, that is, the probability that A occurs when B occurs.

Bernhard formula

It is used to find out the probability of an event and what factors caused it.

Well, in the above example, it is known that event A has occurred, and the probability caused by factor B can be obtained by bernhard formula, and so can factor C and factor D. 。

Classical probability p (a) = number of basic events contained in a/total number of basic events.

Geometric probability P(A)= area/total area

Conditional probability p (a | b) = nab/nb = p (ab)/p (b) = number of basic events contained in ab/number of basic events contained in b.

Properties of probability

Attribute 1.p (φ) = 0.

Characteristic 2 (limited additivity). When n events A 1, …, An are incompatible with each other: P(a 1∨). ..∪An)=P(A 1)+...+P(An)。

Real estate 3. For any event, a: p (a) = 1-p (not a).

Property 4. When events A and B satisfy that A is included in B: P(BnA)=P(B)-P(A), p (a) ≤ p (b).

Property 5. For any event a, p (a) ≤ 1.

Real estate 6. For any two events a and b, p (b-a) = p (b)-p (ab).

Attribute 7 (addition formula). For any two events A and B, p(A∪B)= p(A)+p(B)-p(A∪B)(-) First mathematical induction:

Generally speaking, to prove a proposition related to a positive integer n, there are the following steps:

(1) proves that the proposition holds when n takes the first value.

(2) Assuming that n = k (the first value of k ≥ n, and k is a natural number), it is proved that the proposition is also true when n=k+ 1.

(2) the second mathematical induction:

The second principle of mathematical induction is that there is a proposition related to the natural number n, if:

(1) When n= 1, the proposition holds;

(2) Suppose that the proposition holds when n≤k and when n=k+ 1.

Then, this proposition holds for all natural numbers n.

(3) Spiral induction:

Spiral induction is a variant of induction, and its structure is as follows:

Pi and Qi are two sets of propositions, if:

P 1 holds.

Pi established => all established.

Then π and qi hold true for all natural numbers I.

It is easy to prove that spiral induction is correct by the first mathematical induction.