Math 201 Chapter 4: Probability

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Math 201 Chapter 4 Probability

• Study of Randomness and
• Uncertainty

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Last Time Statistical Inference

• Parameters vs. Statistics
• Sampling distributions
• Unbiased samples
• Sampling variability
• Sampling from large populations

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4.1 Randomness

• Probability is the science of chance behavior
• Chance behavior is unpredictable in the short run
  but has a regular and predictable pattern in the
  long run
• this is why we can use probability to gain useful
  results from random samples and randomized
  comparative experiments
• Applet

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Relative-Frequency Probabilities
Coin flipping What is the probability that it
will land heads?
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Relative-Frequency Probabilities
Toss a thumbtack ? What is the probability that
it will land point down?
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Randomness and Probability

• Random individual outcomes are uncertain but
  there is a regular distribution of outcomes in a
  large number of repetitions
• NoteRandom does not mean haphazard
• 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.

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Probability Rules

• Probability is the long-run proportion of
  repetitions on which an event occurs.
• P(A)measures the likelihood that A will occur
• Probability of event A

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Relative-Frequency Probabilities

• Can be determined (or checked) by observing a
  long series of independent trials
• experience with many samples
• Short runs give only rough estimates
• Independent Trials
• Outcomes of one trial does not influence the
  other trial

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4.2 Probability Models

• Need to model randomness.
• Example
• Toss a coin once ? outcome unknown in advance.
• But what do we know?
• Probability models?
• List all outcomes
• Probability for each outcome

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Sample Spaces

• Sample Space (S) is the list of all possible
  outcomes
• Example Toss a coin
• SHead, Tail
• Toss 4 coins count of Heads.
• S?

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Sample Spaces

• Example Generate a random number between 0 and
  1.
• S?
• Pick a students from class. Ask whether they are
  going home in summer?
• S?
• Pick Two students ?
• SYY,YN,YU,NY, NN,NU,UY,UN,UU

3 outcomes
9 outcomes
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Sample Space Can Contain Infinite No. Of Outcomes
Too!

• Example
• Examine light bulbs coming off an assembly line
  till we observe a defective
• SB, GB, GGB,GGGB,

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Event

• Is an outcome or set of outcomes.
• Example Going home for summer ctd.
  SYY,YN,YU,NY, NN,NU,UY,UN,UU
• Let A be the Event that Both students are
  undecided AUU
• Let B be the Event that at least one of the
  students are undecided
• B YU, NU, UU, UY, UN

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Probability Rule 1

• 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!).

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Probability Rule 2

• 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 3

• The probability that an event does not occur is
  1 minus the probability that the event does
  occur.
• Examples
• 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.
• If the probability that a flight will be on time
  is 0.70, then the probability it will be late is
  0.30.

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Disjoint Events

• Also known as Mutually Exclusive (ME) events
• A and B have no outcomes in common.
• Example Voting

A
B
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Probability Rule 4

• If two events have no outcomes in
  common(disjoint), the probability that one or the
  other occurs is the sum of the two probabilities
• Example
• Age of woman at first child birth
• under 20 25
• 20-24 33
• What is the probability that a woman will be 24
  or younger at first child birth?
• What is the probability that a woman will be over
  24 at first child birth?

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Complement of Event A

• AcOutcomes that are not in A

Ac
A
The event A or B

• Outcomes that are in A or in B
• or
• in BOTH

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• The event A and B
• Outcomes that are in both A and B.
• A and B

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Homework

• 4.5, 4.11, 4.13, 4.17