Unit 4-Lecure 2: Conditional Probability

1 Conditional Probility

1.1 Definition

Let X and Y be events where Y has positive probability, then

Pr[X|Y]::=Pr[X∩Y]Pr[Y]

1.2 The Four Step Method for Conditional Probability

Find the sample spaceDefine event of intrestDetemine outcome probabilitiesCompute event probabilities

1.3 Conditional Probability Product Rule

2 events: Pr[E1∩E2]=Pr[E1]⋅Pr[E2|E1] 3 events: Pr[E1∩E2∩E3]=Pr[E1]⋅Pr[E2|E1]⋅Pr[E3|E1∩E2]

2 The Law of Total Probability

If E1 , E2 , E3 are disjoint and P[E1∪E2∪E3]=1 , then:

Pr[A]=Pr[A|E1]⋅Pr[E1]+Pr[A|E2]⋅Pr[E2]Pr[A|E3]⋅Pr[E3]

3 Bayes’ Rule

Pr[A|B]=Pr[A∩B]Pr[B]

4 Pilosphy of Probability

In a Bayesian approach, a probability is interpreted as a degree of belief in a proposition.In the frequentist view, the probability of an event represents the fraction of trials in which the event occurred

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Reference

[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.