Thinking About Chance

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Chapter 17

• Thinking About Chance

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Thought Question 1

• Here are two very different queries about
  probability
  
• If you flip a coin and do it fairly, what is the
  probability that it will land with heads up?
  
• What is the probability that you will eventually
  own a home, i.e. how likely do you think it is?
  (If you already own a home, what is the
  probability that you will own a different home in
  the next five years?)
• For which question was it easier to provide a
  precise answer? Why?

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Thought Question 2

• Which of the following more closely describes
  what it means to say that the probability of a
  tossed coin landing with heads up is 1/2?
  
• After more and more tosses, the fraction of heads
  will get closer and closer to 1/2.
  
• The number of heads will always be about half of
  the number of tosses.

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Thought Question 3
Explain what is wrong with the following
statement, given by a student as a partial answer
to Thought Question 1 The probability that I wi
ll eventually own a home, or of any other
particular event happening, is 1/2 because either
it will happen or it wont.
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Thought Question 4
Suppose a news article claimed that drinking
coffee doubled your risk of developing a certain
disease. Assume the statistic was based on
legitimate, well-conducted research. What
additional information would you want about the
risk before deciding whether or not to quit
drinking coffee? (Hint Does this statistic provi
de any information on your actual risk?)
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Thought Question 5
A recent study estimated that the relative risk
of developing lung cancer if a woman smoked was
27.9. What do you think is meant by the term
relative risk?
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Two Concepts of Probability

• Personal-Probability Interpretation
• The degree to which a given individual believes
  the event in question will happen.
  
• Personal belief
• Relative-Frequency Interpretation
• The proportion of time the event in question
  occurs over the long run.
  
• Long-run relative frequency

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Relative-Frequency Probabilities

• Two ways to determine
• Physical assumptions (theoretical mathematical
  model)
  
• Repeated observations (empirical results)
• Experience with many samples
• Simulation

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Relative-Frequency Probabilities Summary

• Can be applied when the situation can be repeated
  numerous times (conceptually) and the outcome can
  be observed each time.
  
• Relative frequency (proportion of occurrences) of
  an outcome settles down to one value over the
  long run. That one value is then defined to be
  the probability of that outcome.
• The probability cannot be used to determine
  whether or not the outcome will occur on a single
  occasion (it is a long-run phenomenon).

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Personal or Relative Frequency Probabilities?

• The probability that a lottery ticket will be a
  winner.
  
• The probability that you will get a B in this
  course.
  
• The probability that a randomly selected student
  in one of your professors classes will get a B.
  
• The probability that the 7 a.m. flight from San
  Francisco to New York will be on time on a
  randomly selected day.
  
• The probability that the Atlanta Braves
  professional baseball team will win the World
  Series in the year 2010.

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Risk and Relative RiskCase Study
The following table gives results for whether or
not subjects were still smoking when given a
nicotine patch or a placebo
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Relative Risk

• Risk of continuing to smoke
• Nicotine .533 (just the proportion from the
  table)
  
• Placebo .800
• Relative risk of continuing to smoke when using
  the placebo patch compared with when using the
  nicotine patch is 1.5 ( .800/.533).
  
• The risk of continuing to smoke when using the
  placebo patch is 1.5 times the risk when using
  the nicotine patch.

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Misleading Statistics about Risk

• The baseline is missing.
• The reported risk is not necessarily your risk.

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Case StudyBaseline Risk is Missing

• Premature-birth Risk Found Higher for Teens
  (reported in the Sacramento Bee, April 27, 1995,
  p. A7)
  
• The youngest girls in the study , those aged
  13 to 17, were 90 percent more likely than the
  women in their early 20s to deliver
  prematurely.
• The relative risk was 1.9.
• But what is the absolute risk for women in their
  20s which is used as the baseline?

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Case StudyReported Risk May Not Be Your Risk

• Premature-birth Risk Found Higher for Teens
  (reported in the Sacramento Bee, April 27, 1995,
  p. A7)
  
• The greater risk may be due to lack of support
  from the father rather than the age of the girl.
  
• If you are a pregnant teenage girl with plenty of
  support from the father, this risk may not apply
  to you.

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Key Concepts

• Personal probability
• Long-run Relative Frequency interpretation of
  probability
  
• Relative Risk and cautious interpretation