Section 5.1 - Constructing Models of Random Behavior

P1. Suppose Jack and Jill use a sample of four people who can’t tell the difference between tap water and bottled water. This is the same as flipping four fair coins

a. Construct the probability distribution for the number of people in the sample who would choose the tap water just by chance.

a. 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

Section 5.1 - Constructing Models of Random Behavior

P1. Suppose Jack and Jill use a sample of four people who can’t tell the difference between tap water and bottled water. This is the same as flipping four fair coins

b. What is the probability that all four people will identify the tap water correctly?

c. Is four people a large enough sample to ease Jack’s concern about the reputation of Downhill Research?

Section 5.1 - Constructing Models of Random Behavior

P1. Suppose Jack and Jill use a sample of four people who can’t tell the difference between tap water and bottled water. This is the same as flipping four fair coins

b. 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

c. Is four people a large enough sample to ease Jack’s 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%)

Section 5.1 - Constructing Models of Random Behavior

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

Display 5.7 Forecast: 30ºF

Actual Low Temperature Frequency

20 2

25 8

30 13

35 3

40 1

Section 5.1 - Constructing Models of Random Behavior

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

Section 5.1 - Constructing Models of Random Behavior

Display 5.7 Forecast: 30ºF

Actual Low Temperature Frequency

20 2

25 8

30 13

35 3

40 1

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

Of the 27 days listed on which the NWS forecasted a low of 30°F, 13 days actually had a low of 30°F. The best estimate of the probability is 13/27 ≈ 0.48

Section 5.1 - Constructing Models of Random Behavior

Display 5.7 Forecast: 30ºF

Actual Low Temperature Frequency

20 2

25 8

30 13

35 3

40 1

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

Section 5.1 - Constructing Models of Random Behavior

Display 5.7 Forecast: 30ºF

Actual Low Temperature Frequency

20 2

25 8

30 13

35 3

40 1

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

Section 5.1 - Constructing Models of Random Behavior

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

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

Section 5.1 - Constructing Models of Random Behavior

Display 5.7 Forecast: 30ºF

Actual Low Temperature Probability

20 2/27

25 8/27

30 13/27

35 3/27

40 1/27

c. 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.

Section 5.1 - Constructing Models of Random Behavior

Display 5.7 Forecast: 30ºF

Actual Low Temperature Probability

20 2/27

25 8/27

30 13/27

35 3/27

40 1/27

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.

a. List a sample space that has outcomes that are disjoint and complete

a. Are all outcomes in your sample space equally likely?

a. What is the probability that you get heads and a 3?

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.

a. List a sample space that has outcomes that are disjoint and complete

{H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}

a. Are all outcomes in your sample space equally likely?

Yes

a. What is the probability that you get heads and a 3?

P(H3) = 1/12

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.

a. List a sample space that has outcomes that are disjoint and complete

a. Are all outcomes in your sample space equally likely?

b. 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?

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.

a. 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)

a. 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

Section 5.1 - Constructing Models of Random Behavior

P5. Jean d’Alembert 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 d’Alembert correct about the probability of getting

heads at least once?

Section 5.1 - Constructing Models of Random Behavior

P5. Jean d’Alembert 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:

1. Heads on the first flip

2. Heads on the second flip

3. Heads on neither flip

1. Is this list of outcomes complete? Yes.

2. Are the outcomes disjoint? No: outcome 1 and outcome 2 can both occur in two flips of a fair coin.

3. Are the three outcomes equally likely? No: P(outcome 1) = 1/2; P(outcome 2) = 1/2; P(outcome 3) = 1/4.

4. Is d’Alembert 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.

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.

a. Can you list a sample space?

b. Can you determine the probability that all four students are right-handed?

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.

a. Can you list a sample space?

Yes:

{RRRR RRRL RRLR RLRR LRRR LLLR LLRL LRLL RLLL RRLL LLRR RLRL LRLR RLLR LRRL LLLL}

b. 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.)

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.

a. Show all possible outcomes on

a tree diagram.

b. How many possible outcomes

are there?

c. Are all the outcomes equally

likely?

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.

a. Show all possible outcomes on

a tree diagram.

b. How many possible outcomes

are there? Six• Are all the outcomes equally

likely? It is impossible to tell,

but unlikely.

PriceFlavor

Good

OK

Poor

acceptable

unacceptable

acceptable

acceptable

unacceptable

unacceptable

Section 5.1 - Constructing Models of Random Behavior

P9. A dental clinic has three dentists and seven dental hygienists.

a. If you are assigned a dentist and a dental hygienist at random, how many different pairs can you end up with?

b. What is the probability that you get your favorite dentist and your favorite dental hygienist?

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

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

Section 5.1 - Constructing Models of Random Behavior

P9. A dental clinic has three dentists and seven dental hygienists.

a. 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

b. What is the probability that you get your favorite dentist and your favorite dental hygienist? P(favorite pair) = 1/21

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

Section 5.1 - Constructing Models of Random Behavior

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

Dentist

A B C

Hygienist

a aA aB aC

b bA bB bC

c cA cB cC

d dA dB dC

e eA eB eC

f fA fB fC

g gA gB gC

Section 5.1 - Constructing Models of Random Behavior

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

Aa B

CA

b BCA

c BCA

d BCA

e BCA

f BCA

g BC

Hygienist Dentist