What is Bayes Theorem?
Under the condition of limited information and conditions, based on past data, it helps us to predict the near-real probability of events step by step through dynamic adjustment.
Its basic idea is posterior probability = prior probability * adjustment factor, in which prior probability is subjective probability prediction made under the condition of incomplete information; Adjustment factor is the adjustment of prior probability in the process of continuous improvement of information collection; Posterior probability is the final probability prediction after adjustment.
Conditional probability/total probability/inverse probability
Conditional probability: barrel1p (b | a) = 30/(30+10) = 75% probability of drawing a white ball; The probability P(B|C)=20/(20+20)=50% of drawing white balls from the 2nd barrel, which are all conditional probabilities.
Total probability: the probability of drawing a ball as a white ball is called total probability, which is the sum of the probabilities of 1 barrel drawing a white ball and two barrels drawing a white ball, that is, p (b) = p (b | a) p (a)+p (b | c) p (c) = 75% * 50.
Inverse Probability/Bayesian Theorem: What is the probability P(A|B) of the white ball coming out of the bucket 1? This is a typical problem solved by Bayesian theorem.
How to use Bayesian theorem?
1. Find P(A) first, that is, the probability of grabbing a ball and the ball comes from bucket A, which is called prior probability, that is, the probability that event A is unconstrained (the constraint is to draw a white ball), which is well calculated as 50%.
2. Find P(B|A)/P(B) again, which is called possibility function or adjustment function, that is, adjust the factor of P(A) when catching a white ball under known conditions. According to the above calculations, P(B|A)= 75% and P(B)= 62.5%, and the adjustment coefficient is 75%/62.5%= 1.2.
3. Finally, find P(A|B), also known as posterior probability = prior probability * adjustment function = p (a) * (p (b | a)/p (b)) = 50% *1.2 = 60%.
That is to say, if a ball is drawn, the probability that the ball comes from the bucket 1 is 50% under the condition of incomplete information; When we know that the ball is white, the probability of the ball coming from the barrel 1 will increase by 20% (the adjustment coefficient is 1.2), and the probability of the ball coming from the barrel 1 will increase to 60%.