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Bayes’ Rule: The Monty Hall Problem

Bayes’ Rule: The Monty Hall Problem. Josh Katzenstein Kerry Braxton-Andrew Problem From: http://en.wikipedia.org/wiki/Bayes'_theorem. Image from: http://www.grand-illusions.com/monty2.htm. The Problem:. You are given a choice between three doors: Red, Green and Blue

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Bayes’ Rule: The Monty Hall Problem

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  1. Bayes’ Rule: The Monty Hall Problem Josh Katzenstein Kerry Braxton-Andrew Problem From: http://en.wikipedia.org/wiki/Bayes'_theorem Image from: http://www.grand-illusions.com/monty2.htm

  2. The Problem: • You are given a choice between three doors: Red, Green and Blue • Behind one is a car, behind the other two are goats • You pick the red door Photos from: www.istockphoto.com, www.flickr.com and www.commissionersam.com

  3. The Problem: • The host knows where the car is, but is supposed to open a door without the prize • The host opens the green door, and a goat is behind it • You are given the option to change your choice to blue • What is the probability that the car is behind the blue door given that the presenter chose to open green? Photo from: library.thinkquest.com

  4. The Method • Use Bayes’ Rule to solve • Bayes’ rule says: • Where: • P(PR|G) = Probability of prize in red if host picks green • P(G|PR) = Probability of host picking green if prize is in red • P(PR) = Probability of prize in red • P(G) = Probability of host picking green

  5. The Setup • The probability of the car being in the red door P(PR) is 1/3 • This is the same as the probability of the car being behind any other door • Similar relationships can be set up for the other probabilities

  6. The Solution • Assuming the host has no bias, the probabilities break down like this • If the car is behind the red door, then the host can pick blue or green to open: P(G|PR)=P(B|PR)=1/2 • If the car is behind the green door, then the host would have to pick the blue door: P(G|PG)=0 • If the car is behind the blue door, then the host would have to pick the green door: P(G|PB)=1 • P(G) = 1/3 * 1 + 1/3 * 0 + 1/3 * 1/2 = 1/2

  7. The Odds

  8. So! Where’s the Car? • Looking at the odds • Behind the red door still: 1/3 • Behind the green door: 0 • Behind the blue door: 2/3 • SO…ODDS SAY TO SWITCH YOUR PICK EVERY TIME But what if…….. Image from: johnfenzel.typepad.com

  9. A Bias • Say you’ve watched this show before, you see that the host hates MSU, so rarely picks the green door unless he has to (only 10% of the time) • What does this do to your odds???

  10. Reevaluating the Odds • This changes P(G) • P(G)=1/3*1+1/3*0+1/3*1/10=11/30 • The first term is if he has to • The second term is if he can’t (car is behind green) • The third is if he has a choice, but will only pick green one in ten times

  11. New Odds

  12. So! Where’s the Car Now? • The same place! However, rather than the odds being 2/3 to 1/3 they are now 10/11 to 1/11. • So what would it have meant if the host had picked blue? What were the odds then? • That’s an exercise for you!

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