Probability Models

1
Chapter 18

• Probability Models

2
Probability Rules

• Four simple, logical rules which apply to how
  probabilities relate to each other and to real
  events.
• Review the rules at the beginning of Chapter 18.
• Review the examples presented in Chapter 18!!

3
Probability Rule A

• Any probability is a number between 0 and 1.
• A probability can be interpreted as the
  proportion of times that a certain event can be
  expected to occur.
• If the probability of an event is more than 1,
  then it will occur more than 100 of the time
  (Impossible!).

4
Probability Rule B

• All possible outcomes together must have
  probability 1.
• Because some outcome must occur on every trial,
  the sum of the probabilities for all possible
  outcomes must be exactly one.
• If the sum of all of the probabilities is less
  than one or greater than one, then the resulting
  probability model will be incoherent.

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Probability Rule C

• The probability that an event does not occur is
  1 minus the probability that the event does
  occur.
• As a jury member, you assess the probability that
  the defendant is guilty to be 0.80. Thus you
  must also believe the probability the defendant
  is not guilty is 0.20 in order to be coherent
  (consistent with yourself).
• If the probability that a flight will be on time
  is .70, then the probability it will be late is
  .30.

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Probability Rule D

• If two events have no outcomes in common, they
  are said to be mutually exclusive. The
  probability that one or the other of two mutually
  exclusive events occurs is the sum of their
  individual probabilities.
• Age of woman at first child birth
• under 20 25
• 20-24 33
• 25 ?

24 or younger 58
Rule C 42
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Avoid Being Inconsistent

• If the ways in which one event can occur are a
  subset of those in which another event can occur,
  then the probability of the first event cannot be
  higher than the probability of the one for which
  it is a subset.
• Suppose you see an elderly couple and you think
  the probability that they are married is 80.
• Suppose you think the probability that the
  elderly couple is married with children is 95.
• These two personal probabilities are not
  coherent. Why?

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Avoid Being Inconsistent
Probability of married with children must not be
greater than the probability that the couple is
married.
9
Sampling Distribution

• Tells what values a statistic (calculated sample
  value) takes and how often it takes those values
  in repeated sampling.
• Assigns probabilities to the values a statistic
  can take. These probabilities must satisfy Rules
  A-D.
• Probabilities are often assigned to intervals of
  outcomes by using areas under density curves.
• often this density curve is a normal curve
• can use 68-95-99.7 rule or get probabilities
  from Table B
• sample proportions (p-hats) follow a normal curve

10
Sampling Distribution for Proportion Who Voted
World Almanac and Book of Facts (1995),
Famighetti, R. editor,Mahwah, N.J. Funk and
Wagnalls
56 of registered voters actually voted in the
1992 presidential election. In a random sample of
1600 voters, the proportion who claimed to have
voted was .58.
What is the probability of observing a sample
proportion as large or larger than .58?
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Sampling Distribution for Proportion Who Voted

• If we convert the observed value of .58 to a
  standardized score, we get
• standardized score
• (observed value - mean) / (std dev)
• (.58 - .56) / .012 1.67
• From Table B, this is about the 95th percentile,
  so the probability of observing a value as small
  as .58 is about .95.
• By Rule C (or B), the probability of observing a
  value as large or larger than .58 is 1-.95 .05 .

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Key Concepts

• Rules for probability
• Avoid Inconsistencies
• Sampling Distributions