Note that if A and B are independent then PBAPB Conditional probability

# Note that if a and b are independent then pbapb

• Notes
• 38

This preview shows page 17 - 26 out of 38 pages.

Note that if A and B are independent, then P(B|A)=P(B) Conditional probability: 𝑃𝑃 𝐵𝐵 𝐴𝐴 = 𝑃𝑃 ( 𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵 ) 𝑃𝑃 ( 𝐴𝐴 )

Subscribe to view the full document.

Conditional Probability and the Prosecutor Fallacy P(guilty and fits description) = P(fits description| guilty) * p(guilty) = P(fits description | guilty) = 1 P(guilty) = 1/ 10.000.000 P(fits description) = 0.000002 Hence: P(guilty | fit description) = 1∗ ( 1 / 10 . 000 . 000 ) 0 . 00002 = 1/20
So, this is a nice theoretical example? NO! The Lucia de Berk case in the Netherlands (2003) Paediatric Nurse Life imprisonment for 7 murders and 3 attempted nurse Exonerated in 2010 The Sally Clark case in the UK (1999) Mother of 2 sons Life imprisonment for murdering her sons Sudden death syndrome Exonerated in 2003 Further watching:

Subscribe to view the full document.

Bayes’ rule Suppose you know P(A), P(B|A) and 𝑃𝑃 𝐵𝐵 𝐴𝐴 𝑐𝑐 and you want to calculate P(A|B) (see textbook, example 5.10). Then the definition of conditional probability gives: P A B = 𝑃𝑃 𝐵𝐵 𝐴𝐴 ∗𝑃𝑃 ( 𝐴𝐴 ) 𝑃𝑃 ( 𝐵𝐵 ) And 𝑃𝑃 𝐵𝐵 = 𝑃𝑃 𝐵𝐵 𝐴𝐴 ∗ 𝑃𝑃 𝐴𝐴 + 𝑃𝑃 𝐵𝐵 𝐴𝐴 𝑐𝑐 ∗ 𝑃𝑃 𝐴𝐴 𝑐𝑐 Inserting P(B) in the denominator gives us Bayes’ rule: 𝑃𝑃 𝐵𝐵 𝐴𝐴 ∗ 𝑃𝑃 ( 𝐴𝐴 ) 𝑃𝑃 𝐴𝐴 ∗ 𝑃𝑃 𝐵𝐵 𝐴𝐴 + 𝑃𝑃 𝐴𝐴 𝑐𝑐 ∗ 𝑃𝑃 𝐵𝐵 𝐴𝐴 𝑐𝑐
Let’s play a game! Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the other two a goat.

Subscribe to view the full document.

Win a car or a goat Let us actually play this game! - illusions.com/simulator/montysim.htm
The Monty Hall Problem (2) You pick a door, say No.1, and the host (His name is Monty Hall) , who knows what's behind the doors, opens another door, say No.3, which has a goat. Monty then asks: "Do you want to pick door No.2?" Is it to your advantage to switch your choice?

Subscribe to view the full document.

The Monty Hall Problem (3) It is always the best strategy to switch doors! The probability of winning the car without change of doors is 1/3 The probability of winning the car with change of cars is 2/3. To some people the outcome may be counter-intuitive. To those I have two suggestions: 1. Rephrase the problem such that there are 100 doors and the quiz master opens up 98 after you have chosen a door. Would you change doors then? 2. Let us apply the rules of probability to show that you always have to change doors when you are given the option to change!
The Monty Hall Problem (4) To show that you have to switch doors, suppose that: you pick door #1, call it: “a”. Monty Hall opens door #2, call it “b”. There is one door left (#3): call it “c”. Let A, B, and C be the events that the car is behind door a, b, or c, respectively.

Subscribe to view the full document.

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes