• Multiplication Rule: P(AandB)=P(B)⋅P(A∣B)
• Conditional Probability: P(B∣A)=P(A)P(AandB)
• Law of Total Probability: P(A)=P(B1)P(A∣B1)+P(B2)P(A∣B2)+⋯+P(Bn)P(A∣Bn)
Combining all these equations we get Bayes’ Rule: P(B∣A)=P(A)P(AandB)=P(A)P(B)⋅P(A∣B) =P(B1)P(A∣B1)+P(B2)P(A∣B2)+⋯+P(Bn)P(A∣Bn)P(B)⋅P(A∣B)
I am going to ask my boss to be my reference after applying to another job. If she gives me a good recommendation there is a 0.75 probability that I will get the job. While if she gives me a bad recommendation there is only a 0.25 probability that I will get the job. There is a 60% chance she will give me a good reference and a 40% chance she will give me a bad reference.
I have 3 bags that each contains 5 marbles.
I roll a fair die to decide which bag I will draw from. If I roll a 1,2,3 I will draw a marble from Bag A. If I roll a 4,5 I will draw from Bag B. And if I roll a 6, then I will draw a marble from bag C.
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