random process: we know what outcomes could happen, but we don’t know which particular outcome will happen

two ways to interpret probability

  • frequentist interpretation
  • bayesian interpretation

law of large numbers

disjoint events(mutually exclusive) cannot happen at the same time. p non-disjoint events

for disjoint events A and B, union of disjoint events P(A or B)=P(A)+P(B)

union of non-disjoint events P(J or red) = P(J)+P(red)-P(J and red)

general addition rule: P(A or B) = P(A) +P(B)-P(A and B)

sample space a collection of all possible outcomes of a trial

a probability distribution lists all possible outcomes in the sample space and the probabilities with which they occur

rules of probabilities distribution

  1. the events listed must be disjoint(mutually exclusive)
  2. each 0–1
  3. sum to 1 to represent the entire sample space

complementary events: two disjoint event which sum up to 1.

disjoint -> complementary

two precesses are independent if knowing the outcome of one provides no useful information about the outcome of the other.

knowing B tells nothings about A P(A|B) = P(A) where A and B are independent.

product rule for independent events: P(A and B) = P(A)*P(B)

when choose sample, random select means the samples are independent

marginal probability joint probability conditional p

calculate condition p based on the bayed’s theorem:

bayes inference

bayes inference



\sigma = \sqrt{np(1-p)}





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