Chapter 11 – Understanding Randomness
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Pick one of the numbers at random and
write it down. Don’t shown anyone.
I chose number ________.
August 24th, 2015 Results
Chapter 11 – Understanding Randomness
5% of all people pick the number 1
75% of all people pick the number 3
20% of all people pick either 2 or 4
• Humans do not do well at just picking something at
random.
Chapter 11 – Calculator Skills
Random Integer
MATH
Choose ‘PRB’ tab
Choose ‘5: randInt(‘
4.
RANDINT(Low , High , How many)
RANDINT(1,100,5)
will produce five random integers
between 1 and 100 inclusive.
Chapter 11 – Calculator Skills
5. Every TI-84 will produce the same
pseudorandom integers if the calculator
starts at the same place in the calculator’s
random number table.
• To “seed” the calculator, means to tell
the calculator where to start on it’s
random number table.
Chapter 11 – Calculator Skills
Seed the Random Number Generator
1) Enter a number
2) STO>
3) MATH
Choose ‘PRB’ tab
Choose ‘1: rand‘
“Seed”: 0  rand
Chapter 11 – Calculator Skills
Seed the Random Number Generator
“Seed”: 0  rand
Generate 5 random numbers 1 to 100:
RANDINT(1, 100, 5)
Results are:
95 91 15 52 41
These are only Pseudorandom numbers. Because
the outcomes were predictable.
Chapter 11 – Understanding Randomness
8. Thirty-eight percent of the people in the
United States have type O positive blood. Three
unrelated people in the U.S. are selected at
random. Design a simulation to determine how
often all three will have type O positive blood.
Chapter 11 – Understanding Randomness
8. Thirty-eight percent of the people in the United States have
type O positive blood. Three unrelated people in the U.S. are
selected at random. Design a simulation to determine how
often all three will have type O positive blood.
a. Identify the component to be repeated.
Component: Check the blood type of 1 person.
Chapter 11 – Understanding Randomness
8. Thirty-eight percent of the people in the United States
have type O positive blood. Three unrelated people in the
U.S. are selected at random. Design a simulation to determine
how often all three will have type O positive blood.
a. Identify the component to be repeated.
Component: Check the blood type of 1 person.
b. Explain how you will model the outcome.
Let 00 – 37 represent the person selected has type
O positive blood.
Let 38 – 99 represent the person selected doesn’t
have type O positive blood.
Chapter 11 – Understanding Randomness
8. Thirty-eight percent of the people in the United States have
type O positive blood. Three unrelated people in the U.S. are
selected at random. Design a simulation to determine how
often all three will have type O positive blood.
c. Explain how you will simulate the trial.
Each trial consists of identifying 3 pair of digits as
Y(Person has O positive) or N(Person Doesn’t have O
positive). Each pair represents one person. We want 3
persons.
d. Clearly state the response variable.
The response variable is whether or not all three
persons have type O positive blood type.
8. Thirty-eight percent of the people in the United States have type O
positive blood. Three unrelated people in the U.S. are selected at
random. Design a simulation to determine how often all three will have
type O positive blood.
e. Run several trials. We’ll run 8 trials using the
following random numbers:
Y
N N Y N N N Y N Y
Trial #
1
2
3
4
30/Y
15/Y
72/N
12/Y
:
:
7
29/Y
Y N
Outcomes
73/N 47/N
71/N 83/N
27/Y 97/N
25/Y
77/N
:
28/Y
:
31/Y
All O+?
NO
NO
NO
NO
:
YES
5
NO
6
NO
8
YES
Chapter 11 – Understanding Randomness
8. Thirty-eight percent of the people in the United States have
type O positive blood. Three unrelated people in the U.S. are
selected at random. Design a simulation to determine how
often all three will have type O positive blood.
f. Analyze the response variable.
2 / 8 or 25% of the trials found all three were O+
g. State your Conclusion in context.
Our simulation showed that if three unrelated
people in the U.S. were chosen at random, all three
are O+ about 25% of the time. However, it should be
noted that only 8 trials were run.
Chapter 11 – Understanding Randomness
9. Thirty-eight people out of 100 in the United States
have type O positive blood. Design a simulation to
determine how many people on average would need
to be selected at random to get one person that has
type O positive blood.
Chapter 11 – Understanding Randomness
9. Thirty-eight people out of 100 in the United States have
type O positive blood. Design a simulation to determine how
many people on average would need to be selected at
random to get one person that has type O positive blood.
a. Identify the component to be repeated.
Component: Check the blood type of 1 person
b. Explain how you will model the outcome.
Let 01 – 38 represent that the person selected has
type O positive blood.
Let 39 – 99 & 00 represent that the person selected
doesn’t have type O positive blood.
Chapter 11 – Understanding Randomness
9. Thirty-eight people out of 100 in the United States have type O positive blood.
Design a simulation to determine how many people on average would need to be
selected at random to get one person that has type O positive blood.
c. Explain how you will simulate the trial.
Each trial consists of identifying a pair of digits as
N(Person doesn’t have O positive) or Y(Person has O
positive). Each pair represent one person. We will
keep checking until a person has been identified as
having O+ blood.
d. Clearly state the Response variable.
The response variable is how many individuals were
checked to find the first person with type O positive
blood.
9. Thirty-eight people out of 100 in the United States have type O positive blood.
Design a simulation to determine how many people on average would need to be
selected at random to get one person that has type O positive blood.
e. Run several trials. We’ll run 12 trials using the
following random numbers:
Trial #
1
2
3
4
5
# of People
Outcomes
95/N
30/Y
91/N
10/Y
34/Y
83/N
62/N
25/Y
4
1
78/N
58/N
02/Y
4
1
1
Chapter 11 – Understanding Randomness
9. Thirty-eight people out of 100 in the United States have type O positive blood.
Design a simulation to determine how many people on average would need to be
selected at random to get one person that has type O positive blood.
f. Analyze the response variable.
4+1+4+1+1+1+1+1+1+1+4+4 = 24
24/12 = 2
g. State your Conclusion in context.
Our simulation showed that if unrelated people in
the U.S. were chosen at random that on average 2
people would need to be selected to get a person
with O+ blood. However, it should be noted that only
12 trials were run.
Chapter 11 – Understanding Randomness
10. Design a simulation to determine on
average how many times someone should roll a
pair of dice in order to get “snake eyes” (Two
ones).
a. Identify the Component to be repeated.
Component: Rolling a pair of dice once
The chance of rolling “snake eyes” isn’t the same as
the chance of rolling a seven. Fill-in the table below to
see this.
ONE
ONE
2
TWO 3
THREE 4
FOUR 5
FIVE 6
SIX 7
2: 1/36
7: 6/36
12: 1/36
TWO
3
4
5
6
7
8
3: 2/36
8: 5/36
THREE FOUR
4
5
6
7
8
9
4: 3/36
9: 4/36
5
6
7
8
9
10
FIVE
6
7
8
9
10
11
5: 4/36
10: 3/36
SIX
7
8
9
10
11
12
6: 5/36
11: 2/36
Chapter 11 – Understanding Randomness
10. Design a simulation to determine on average how many
times someone should roll a pair of dice in order to get “snake
eyes” (Two ones).
b. Explain how you will model the outcome.
Let 00 represent rolling “snake eyes”.
Let 01 – 35 represent not rolling “snake eyes”.
Ignore 36 – 99. If a pair for this set is found check
the next pair of digits.
Chapter 11 – Understanding Randomness
10. Design a simulation to determine on average how many times someone should roll
a pair of dice in order to get “snake eyes” (Two ones).
c. Explain how you will simulate the trial.
Each trial consists of identifying a pair of digits as
N(Not “snake eyes”) or Y(“snake eyes”) or I(ignore).
Each pair represents one roll of two dice. If a pair of
digit is to be ignored, then they will not be counted as
a roll of the dice. The trail will end once “snake eyes”
is rolled.
d. Clearly state the response variable.
The response variable is how many times the pair of
dice were rolled in order to get “snake eyes”.
Chapter 11 – Understanding Randomness
10. Design a simulation to determine on average how many times someone could roll
a pair of dice in order to get “snake eyes” (Two ones).
e. Run several trials. We’ll run 2 trials using the
following random numbers:
65358 70469 87149 89509 72176 18103 55169 79954 72002 20582
05409 20831 01911 60767 55248 79253 00317 84120 77772 50103
75642 64510 79185 86109 67056 01991 14620 23598 88515 35696
T
Outcomes
C
#
t
1 I N I N I I N I I N I N I I N I N I I I I Y 8
2 NN I N I I N N N I N N I I N I I I Y
10
Chapter 11 – Understanding Randomness
10. Design a simulation to determine on average how many times someone could roll
a pair of dice in order to get “snake eyes” (Two ones).
f. Analyze the response variable.
8 + 10 = 18 18/2 = 9
h. State your conclusion in context.
From our simulation, on average we expect to roll the
pair of dice nine times in order to get “snake eyes”.
However, it should be noted that only 2 trials were
run.
Chapter 11 – Understanding Randomness
11. How often will someone who completely guesses on a 29 question
exam pass the exam?
a. Identify the Component to be repeated.
Component: Answering one test question
b. Explain how you will model the outcome.
Since there is a 20% chance of guessing correctly…
Let 0 – 1 represent guessing correctly
Let 2 – 9 represent not guessing correctly
Chapter 11 – Understanding Randomness
11. How often will someone who completely guesses on a 29 question
exam pass the exam?
c. Explain how you will simulate the trial.
Each trial consists of identifying a digit as
R( guessed right) or W(guessed wrong). Each
digit represents one question. There are 29 questions,
so stop after checking 29 digits.
d. Clearly state the response variable.
Since 21 right out of 29 is passing, the response
variable is were 21 questions answered correctly?
Chapter 11 – Understanding Randomness
11. How often will someone who completely guesses on a 29 question
exam pass the exam?
e. Run several trials.
You’ve been given a random number table. How many
trails can you run?
f. Analyze the response variable.
How many times did a trial end up with someone
passing?
g. State your conclusion in context.
What would your conclusion be?