1. If A and B are independent, then P(A and B) = P(A) * P(B).

2. Else (in general), P(A and B) = P(A given B) * P(B) (if A and B are not independent, then P(A and B happen at the same time) = P(A happened given that B already happen) * P(B);

3. If A and B are mutually exclusive (e.g., if one happens, the other one can’t),

P(A or B) = P(A) + P(B).

4. Else (in general), P(A or B) = P(A) + P(B) – P(A and B). P(A or B happen) = P(A) + P(B) – P(A and B happen at the same time).

Bayes Therom: P(A|B) = P(B|A)*P(A)/P(B). Introductory example:

Say we have a conversation with someone in a train, where there are equal number of men and women. So we know that the probability of we are having conversation with a woman is 50%. Assume that we had this conversation with someone with long hair, and women tends to have long hair (75%). Question, what is the probability that we are talking to a woman in the train.

Lets assume A: we are talking to a woman. B: we are talking to someone with long hair, then we can interpret the problem of what is P(A|B) : what is the probability that we are talking to a woman given that we already know we are talking to someone with long hair.

According to the Bayes Therom

P(A|B) = P(B|A) * P(A)/P(B). we know that P(B|A) = 75% (75% of the woman has long hair) and P(A) = 50% (half of the people in the train are women). How about P(B)? P(B) = P(B|A)*P(A) + P(B|~A)*P(~A) meaning adding all the man with long hair and all the woman with long hair. (Supposedly P(B|~A) = 15%, 15% of the man has long hair), Then we will kown that our final P(A|B) ~= 83%.

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