Probability and expectation

Probability

Probability axioms:
A function that satisfies the following 3 conditions is called a probability function.
(1) 0<=p (A) <=1 0
(2) P (s) =1 p (s) = 1
(3) If A1a2a3⋯a_1a_2a_3\cdots is a series of 22 unrelated events, i.e. for ∀i≠j,ai⋂aj=∅\forall i \not= j,a_i \bigcap a_j = \emptyset, then
P (⋃k=1∞ak) =∑k=1∞p (Ak) p\left (\bigcup_{k=1}^{\infty}a_k\right) = \sum_{k=1}^{\infty}p (A_k)
(can be added to the list)

Conditional probabilities:
b for an event satisfies P (a) >0 p (a) >0, and for any event B, defines the conditional probability of B about a (the probability of event B occurring under event a occurrence condition)
P (b| A) =p (A⋂B) p (a) p (b| a) = \frac {P (a\bigcap B)}{p (a)}

Independent:
If A, B satisfies P (a⋂b) =p (a) p (b) p (a \bigcap B) = P (a) p (b), which is said to be independent
can be introduced P (A) =p (a| B) p (A) = P (a| B

Probability of finding the result of full probability formula:< >
The B1b2b3⋯bn b_1b_2b_3\cdots B_n is a series of events in the sample space S that are disjoint from each other, and are satisfied with the complete set, that is s=⋃nk=1bk S = \bigcup_{k=1}^n B_k, and