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Bayes' rule
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Bayes' rule
Lessons
Recall:
• 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)
• 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)
- IntroductionDeriving Bayes' rule
- 1.Bayes' Rule
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.a)What is the probability that I will get the job?b)Given that I got the job what is the probability that she gave me a good reference - 2.I have 3 bags that each contains 5 marbles.
Bag A:
Bag B:
Bag C:
2 Green
4 Green
5 Green
3 Red
1 Red
0 Red
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.a)What is the probability that I draw a red marble?b)Suppose that I drew a green marble from a bag. What is the probability that I rolled a 6? - 3.False Positives
A blood test is 95% effective when diagnosing a diseased person. However this blood test also incorrectly diagnoses a healthy person 5% of the time. If 0.1% of the population actually has this disease, then what is the probability that a person has the disease given that they tested positive?
I like this, gives you hope if you get diagnosed for a serious disease
Make note of how many healthy people and how many diseased people get diagnosed. Also make a tree diagram illustrating this.