Section 5.1 - Constructing Models of Random Behavior

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Section 5.1 - Constructing Models of Random
Behavior

• P1. Suppose Jack and Jill use a sample of four
  people who cant tell the difference between tap
  water and bottled water. This is the same as
  flipping four fair coins
• Construct the probability distribution for the
  number of people in the sample who would choose
  the tap water just by chance.

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Section 5.1 - Constructing Models of Random
Behavior

• Construct the probability distribution for the
  number of people in the sample who would choose
  the tap water just by chance.
• Let T tap water, B bottled water. List all
  possible outcomes
• TTTT BBBB
• TTTB TTBT TBTT BTTT BBBT BBTB BTBB TBBB
• TTBB BBTT TBTB BTBT TBBT BTTB

Number Who Choose T Probability
0 1/16
1 4/16
2 6/16
3 4/16
4 1/16
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Section 5.1 - Constructing Models of Random
Behavior

• P1. Suppose Jack and Jill use a sample of four
  people who cant tell the difference between tap
  water and bottled water. This is the same as
  flipping four fair coins
• What is the probability that all four people will
  identify the tap water correctly?
• Is four people a large enough sample to ease
  Jacks concern about the reputation of Downhill
  Research?

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Section 5.1 - Constructing Models of Random
Behavior

• P1. Suppose Jack and Jill use a sample of four
  people who cant tell the difference between tap
  water and bottled water. This is the same as
  flipping four fair coins
• What is the probability that all four people will
  correctly identify the tap water correctly?
• The probability that all four people will guess
  correctly is 1/16, or 0.0625
• Is four people a large enough sample to ease
  Jacks concern about the reputation of Downhill
  Research?
• Probably not. There is a 6.25 chance that all
  four people will guess the correct answer. (Good
  rule of thumb less than 5)

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Section 5.1 - Constructing Models of Random
Behavior

• P2. Display 5.7 gives the actual low temperature
  (to the nearest 5F) in Oklahoma City on days
  when the National Weather Service forecast was
  for a low temperature of 30F.

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Frequency
20 2
25 8
30 13
35 3
40 1
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Section 5.1 - Constructing Models of Random
Behavior

• Suppose the forecast for tomorrow is for a low
  temperature of 30F. What is your estimate of the
  probability that the low temperature really will
  be approximately 30F?

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Frequency
20 2
25 8
30 13
35 3
40 1
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Section 5.1 - Constructing Models of Random
Behavior

• Suppose the forecast for tomorrow is for a low
  temperature of 30F. What is your estimate of the
  probability that the low temperature really will
  be approximately 30F?
• Of the 27 days listed on which the NWS
  forecasted a low of 30F, 13 days actually had a
  low of 30F. The best estimate of the probability
  is 13/27 0.48

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Frequency
20 2
25 8
30 13
35 3
40 1
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Section 5.1 - Constructing Models of Random
Behavior

• Using these data, make a table that gives the
  estimated probability distribution for the actual
  low temperature when the forecast is 30F.

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Frequency
20 2
25 8
30 13
35 3
40 1
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Section 5.1 - Constructing Models of Random
Behavior

• Using these data, make a table that gives the
  estimated probability distribution for the actual
  low temperature when the forecast is 30F.

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Frequency Probability
20 2 2/27
25 8 8/27
30 13 13/27
35 3 3/27
40 1 1/27
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Section 5.1 - Constructing Models of Random
Behavior

• Does the method of forecasting appear to give a
  prediction that tends, on average, to be too
  warm, or too cold?

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Probability
20 2/27
25 8/27
30 13/27
35 3/27
40 1/27
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Section 5.1 - Constructing Models of Random
Behavior

• Does the method of forecasting appear to give a
  prediction that tends, on average, to be too
  warm, or too cold?
• Look at the incorrect predictions. The actual
  temperature was colder than predicted 10 times,
  and warmer than predicted 4 times. The
  forecasting method tends to give predictions that
  are too warm.

Display 5.7 Forecast 30ºF Display 5.7 Forecast 30ºF
Actual Low Temperature Probability
20 2/27
25 8/27
30 13/27
35 3/27
40 1/27
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Section 5.1 - Constructing Models of Random
Behavior

• P3. Suppose you flip a coin and then roll a die.
  If you get heads and a 3, then your outcome is
  H3.
• List a sample space that has outcomes that are
  disjoint and complete
• Are all outcomes in your sample space equally
  likely?
• What is the probability that you get heads and a
  3?

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Section 5.1 - Constructing Models of Random
Behavior

• P3. Suppose you flip a coin and then roll a die.
  If you get heads and a 3, then your outcome is
  H3.
• List a sample space that has outcomes that are
  disjoint and complete
• H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6
• Are all outcomes in your sample space equally
  likely?
• Yes
• What is the probability that you get heads and a
  3?
• P(H3) 1/12

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Section 5.1 - Constructing Models of Random
Behavior

• P4. You randomly choose two workers to be laid
  off from a group of workers ages 28, 35, 41, 47,
  and 55.
• List a sample space that has outcomes that are
  disjoint and complete
• Are all outcomes in your sample space equally
  likely?
• What is the probability that the two youngest
  people are the ones laid off?
• What is the probability that the mean age of
  those laid off is 40 or more?

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Section 5.1 - Constructing Models of Random
Behavior

• P4. You randomly choose two workers to be laid
  off from a group of workers ages 28, 35, 41, 47,
  and 55.
• List a sample space that has outcomes that are
  disjoint and complete
• (28,35) (28,41) (28,47) (28,55) (35,41) (35,47)
  (35,55)
• (41,47) (41,55) (47,55)
• Are all outcomes in your sample space equally
  likely? Yes.
• What is the probability that the two youngest
  people are the ones laid off? P((28,35)) 0.10
• What is the probability that the mean age of
  those laid off is 40 or more?
• Let A (28,55) or (35,47) or (35,55) or (41,47)
  or (41,55) or (47,55)
• P(A) 6/10 0.60

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Section 5.1 - Constructing Models of Random
Behavior

• P5. Jean dAlembert was coauthor of a 35-volume
  Encyclopedie. In it, he wrote that the
  probability of getting heads at least once in two
  flips of a fair coin is 2/3. He said that these
  three outcomes were equally likely
• Heads on the first flip
• Heads on the second flip
• Heads on neither flip
• Is this list of outcomes complete?
• Are the outcomes disjoint?
• Are the three outcomes equally likely?
• Is dAlembert correct about the probability of
  getting heads at least once?

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Section 5.1 - Constructing Models of Random
Behavior

• P5. Jean dAlembert was coauthor of a 35-volume
  Encyclopedie. In it, he wrote that the
  probability of getting heads at least once in two
  flips of a fair coin is 2/3. He said that these
  three outcomes were equally likely
• Heads on the first flip
• Heads on the second flip
• Heads on neither flip
• Is this list of outcomes complete? Yes.
• Are the outcomes disjoint? No outcome 1 and
  outcome 2 can both occur in two flips of a fair
  coin.
• Are the three outcomes equally likely? No
  P(outcome 1) 1/2 P(outcome 2) 1/2 P(outcome
  3) 1/4.
• Is dAlembert correct about the probability of
  getting heads at least once? No
• P(at least one head) P(HH or HT or TH) 3/4
• P(at least one head) 1 - P(TT) 1 - 1/4 3/4.

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Section 5.1 - Constructing Models of Random
Behavior

• P6. Suppose you pick four students at random from
  MOHS and check whether they are left-handed or
  right handed.
• Can you list a sample space?
• Can you determine the probability that all four
  students are right-handed?

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Section 5.1 - Constructing Models of Random
Behavior

• P6. Suppose you pick four students at random from
  MOHS and check whether they are left-handed or
  right handed.
• Can you list a sample space?
• Yes
• RRRR RRRL RRLR RLRR LRRR LLLR LLRL LRLL RLLL
  RRLL LLRR RLRL LRLR RLLR LRRL LLLL
• Can you determine the probability that all four
  students are right-handed?
• No. You need to know the percentage of students
  at MOHS who are right-handed.
• (Right- and Left-handedness are not equally
  likely.)

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Section 5.1 - Constructing Models of Random
Behavior

• P8. Suppose you ask a person to taste a
  particular brand of strawberry ice cream and
  evaluate it as good, okay, or poor on flavor and
  as acceptable or unacceptable on price.
• Show all possible outcomes on
• a tree diagram.
• How many possible outcomes
• are there?
• Are all the outcomes equally
• likely?

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Section 5.1 - Constructing Models of Random
Behavior

• P8. Suppose you ask a person to taste a
  particular brand of strawberry ice cream and
  evaluate it as good, okay, or poor on flavor and
  as acceptable or unacceptable on price.
• Show all possible outcomes on
• a tree diagram.
• How many possible outcomes
• are there? Six
• Are all the outcomes equally
• likely? It is impossible to tell,
• but unlikely.

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Section 5.1 - Constructing Models of Random
Behavior

• P9. A dental clinic has three dentists and seven
  dental hygienists.
• If you are assigned a dentist and a dental
  hygienist at random, how many different pairs can
  you end up with?
• What is the probability that you get your
  favorite dentist and your favorite dental
  hygienist?
• Illustrate your answer in part a with a two-way
  table.
• Illustrate your answer in part a with a tree
  diagram.

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Section 5.1 - Constructing Models of Random
Behavior

• P9. A dental clinic has three dentists and seven
  dental hygienists.
• If you are assigned a dentist and a dental
  hygienist at random, how many different pairs can
  you end up with?
• 3 dentists x 7 hygienists 21 pairs
• What is the probability that you get your
  favorite dentist and your favorite dental
  hygienist? P(favorite pair) 1/21
• Illustrate your answer in part a with a two-way
  table.

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Section 5.1 - Constructing Models of Random
Behavior

1. Illustrate your answer in part a with a two-way
   table.

Dentist Dentist Dentist
A B C
Hygienist a aA aB aC
Hygienist b bA bB bC
Hygienist c cA cB cC
Hygienist d dA dB dC
Hygienist e eA eB eC
Hygienist f fA fB fC
Hygienist g gA gB gC
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Section 5.1 - Constructing Models of Random
Behavior

1. Illustrate your answer in part a with a tree
   diagram.