Probability (Part 2)

M. Drew LaMar
February 12, 2016

“I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive.”

- Plato

Introduction to Biostatistics, Spring 2016

Course Announcements

  • Exam #1: Monday, February 22, 2016
    • Content based - No R!! See Lu's email for more info.
    • Whitlock & Schluter, Chapters 1-5
    • Ruxton & Colegrave, Chapters 1-2
  • Solutions for Homework #2 and 3 are on Blackboard!!!
  • Homework #4 is posted on Blackboard
    • Do NOT do in R-Markdown
    • Do in Microsoft Word, starting with Lu's template for Ruxton & Colegrave
    • Insert pictures/scans of written HW into Word Document
  • Lab #4 - Turn in anything you want in a Word Document

Tips for Solving Probability Problems

  1. Write out the probability that you're are being asked to find. Is it a conditional probability? AND? OR?
  2. Identify probabilities that you are given (again, are these conditionals? ANDs? ORs?)
  3. Draw a probability tree (if appropriate)

Let's Make a Deal

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Let's Make a Deal

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Let's Make a Deal

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Monty Hall Problem

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

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Monty Hall Problem

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Hints

  • What probability do you need to find?
  • What probabilities do you know?
  • Draw a probability tree
  • Each game is a random trial
  • What decisions are made?
  • Make decisions with equal probability

Practice: Contingency tables

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[smoker]?

Answer: 52000/100000 = 0.52

Question: What is Pr[cancer]?

Answer: 9568/100000 = 0.09568

Practice: Contingency tables

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[cancer | smoker]?

Answer: 8944/52000 = 0.172

Question: What is Pr[smoker | cancer]?

Answer: 8944/9568 = 0.9347826

Practice: Contingency tables

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[smoker AND cancer]?

Answer: 8944/100000 = 0.08944

Visualizing probability - Mosaic plots

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Visualizing probability - Probability trees

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Visualizing probability - Probability trees

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Visualizing probability - Probability trees

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Visualizing probability - Probability trees

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Practice Problem #11

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