Download Probability and Statistics Unit 1 Exploring Data: Distributions

Unit 4
Randomness in Data
Topic 14
Probability
(page 303)
OVERVIEW
You have been studying methods for analyzing data,
from displaying them graphically to describing them
verbally and numerically. Earlier, you saw how to
create your own samples of data by sampling from a
larger population. Now we turn our attention to
drawing inferences about the population based on
a sample. As you learned earlier, this inference
process is feasible only if you have randomly
selected the sample from the population.
OVERVIEW
At first glance it might seem that introducing
randomness into the process would make it more
difficult to draw reliable conclusions. Instead,
you will find that randomness produces patterns
that allow us to quantify how close the sample
will come to the population result. This topic
introduces you to the idea of probability and asks
you to explore some of its properties.
Do the Preliminaries
(page 304)
Question number 7 assumes that
you are playing solitaire by the rules.
Essential Question
What are the properties
of probability?
Activity 14-1
Random Babies
(pages 305 to 308)
When it is not feasible to actually carry out an activity
to investigate what would happen in the long run,
simulation
is used instead. Simulation is an
artificial representation of a
random
process
used to study its long-term properties.
Modified Directions!
1. Take 2 sheets of scrap paper.
2. Cut one (1) into 4 equal sections and write a
baby’s first name on each section.
3. Divide, but do NOT cut the second sheet into 4
equal sections and write a mother’s last name in
each section.
4. Shuffle the babies names and randomly place
them on the mother’s names.
5. Record the requested information.
(a)
Did Mrs. Johnson get the right baby?
yes
or no
____________
Did all mothers get the wrong baby?
yes
or no
____________
How many mothers got the right baby?
0 to 4
__________
(b)
1.
2.
3.
Part (a) is repetition #1. Repeat this 4 more times.
Shuffle the babies names and randomly place them
on the mother’s names for each repetition.
Record the requested information.
(b)
Repetition #
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
# of matches
(use 0 to 4)
1
2
3
4
5
(b)
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
Johnson matches?
(c)
student
cum reps
1
5
2
10
3
15
4
20
5
25
6
30
7
35
8
40
9
45
10
50
11
55
12
60
13
65
14
70
15
75
16
80
17
85
18
90
19
95
20
100
of these 5
cum tot
cum prop
All wrong?
of these 5
cum tot
cum prop
Count the number of your Johnson matches!
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
Enter 3 for “Of these 5”.
Johnson matches?
(c)
student
cum reps
of these 5
cum tot
cum prop
1
5
3
3
.60
2
10
3
15
4
20
5
25
6
30
7
35
8
40
9
45
10
50
11
55
12
60
13
65
14
70
15
75
16
80
17
85
18
90
19
95
20
100
All wrong?
of these 5
cum tot
cum prop
Count the number of your “All wrong?”!
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
Enter 2 for “Of these 5”.
Johnson matches?
(c)
All wrong?
student
cum reps
of these 5
cum tot
cum prop
of these 5
cum tot
cum prop
1
5
3
3
.60
2
2
.40
2
10
3
15
4
20
5
25
6
30
7
35
8
40
9
45
10
50
11
55
12
60
13
65
14
70
15
75
16
80
17
85
18
90
19
95
20
100
(d)
(d)
This is just an example of what your graph may look like.
.28
(e) The proportion of trials where Johnson matches appears to be approaching ____.
.33
The proportion with no matches appears to be approaching _____.
The probability of a random event is the
long-run proportion (or relative frequency) of times the
random process were
repeated over and over under identical conditions.
event would occur if the
One can approximate a probability by simulating the
large number of times. Simulation leads to
empirical estimate of the probability.
process a
an
Empirical results are based on experience, observations,
or experimentation rather than theory or pure logic.
(f)
Combine your results with the rest of the class.
Please note that you will be entering 5 tally marks.
# of
matches
0
1
2
3
4 total
count
proportion
1.00
Tally the # of 0 matches from part (b).
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
I will enter 2 tally marks for the two 0 matches.
Enter the 2 tally marks for the # of 0 matches.
# of
matches
0
1
2
3
4 total
||
count
proportion
1.00
Tally the # of 1 matches from part (b).
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
I enter NO tally marks since there are none.
Enter NO tally marks for the # of 1 matches.
# of
matches
0
1
2
3
4 total
||
count
proportion
1.00
Tally the # of 2 matches from part (b).
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
I will enter 2 tally marks for the two 2 matches.
Enter the 2 tally marks for the # of 2 matches.
# of
matches
0
||
1
2
3
4 total
||
count
proportion
1.00
Tally the # of 3 matches from part (b).
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
I enter NO tally marks since there are none.
Enter NO tally marks for the # of 3 matches.
# of
matches
0
||
1
2
3
4 total
||
count
proportion
1.00
Tally the # of 4 matches from part (b).
Repetition #
1
2
3
4
5
Johnson match?
(use Y or N)
All wrong?
(use Y or N)
N
Y
Y
N
Y
Y
N
N
Y
N
0
4
2
0
2
# of matches
(use 0 to 4)
I will enter 1 tally mark for the one 4 matches.
Enter the 1 tally mark for the # of 4 matches.
# of
matches
0
||
1
2
||
3
4 total
|
count
PLEASE TAKE NOTE THAT1.00
5
TALLY MARKS ARE ENTERED!
proportion
(f)
Now enter your data on the board.
Remember, you will be entering 5 tally marks.
# of
matches
0
1
2
3
4 total
count
proportion
1.00
(g) In __________ of these simulated cases, at least one
mother got the correct baby.
(h) My empirical estimate of the probability of no matches
is __________.
(i) My empirical estimate of the probability of at least one
match is __________.
(j) An outcome of exactly 3 matches is impossible because
… if there are 3 matches, then all 4 must match.
NO
(k) Is it impossible to get four matches? _________
YES
Would you call it rare? _________
or No
Would you call it unlikely? Yes
_________
(l) Would you call a result of 0 matches,
NO
or 1 match, or 2 matches, to be unlikely? _________
Essential Question
What are empirical estimates
of probabilities?
Activity 14-2
Random Babies
(continued)
(pages 309 to 311)
In situations where the outcomes of a
random
process are
equally likely, exact probabilities can be calculated by listing all
of the possible outcomes and
that
correspond
counting
the proportion
to the event of interest. The listing of
all possible outcomes is called the
sample space .
Random Babies Sample Space
1234
1243
1324
1342
1423
1432
2134
2143
2314
2341
2413
2431
3124
3142
3214
3241
3412
3421
4123
4132
4213
4231
4312
4321
24 different arrangements for
(a) There are _________
returning the 4 babies to their mothers.
How to get that without all the writing.
There are 4 different babies going to 4 different mothers.
Order really matters!
If you don’t believe me, ask any new mother.
This is a permutation, which is
an arrangement of “r” objects from a set of “n” objects.
nPr
= P(n,r)
! means FACTORIAL.
n!
=
(n - r)!
Permutation
This is an arrangement of
“4” babies from a set of “4” babies.
4!
=
P
=
P(4,4)
4 4
(4 - 4)!
4 × 3× 2 ×1
=
0!
= 24
0! = 1
You don’t
believe me?
Ask your
calculator.
Permutation
To enter this on your calculator,
follow this sequence …
Enter 4 then press…
MATH → PRB ↓ nPr
enter 4 then press ENTER
4 nPr 4
24
For Your Information (FYI)
If order did NOT matter.
This is would be a combination, which is also
an arrangement of “r” objects from a set of “n” objects.
n!
nCr = C(n,r) =
r!(n - r)!
4C4
= 1 … remember order doesn’t matter.
(b)
matches
matches
matches
matches
matches
matches
4
2
2
1
1
2
2
0
1
0
0
1
1
0
2
1
0
0
0
1
1
2
0
0
1234:_______ 1243:_______ 1324:_______ 1342:_______ 1423:_______ 1432:_______
2134:_______ 2143:_______ 2314:_______ 2341:_______ 2413:_______ 2431:_______
3124:_______ 3142:_______ 3214:_______ 3241:_______ 3412:_______ 3421:_______
4123:_______ 4132:_______ 4213:_______ 4231:_______ 4312:_______ 4321:_______
(c)
In how many arrangements is the number of “matches” equal to exactly:
1
0
6
8
9
4:_______ 3:_______ 2:_______ 1:_______ 0:_______
1 + 0 + 6 + 8 + 9 = 24
(c)
In how many arrangements is the number of “matches” equal to exactly:
1
0
6
8
9
4:_______ 3:_______ 2:_______ 1:_______ 0:_______
Divide each number by 24.
(d)
Probabilities for the corresponding numbers:
.04
0
.25
.33
.38
4:_______ 3:_______ 2:_______ 1:_______ 0:_______
[comment]
The exact probabilities should be
fairly close to the simulated results.
An empirical estimate from a simulation
generally gets
closer
to the actual probability
as the number of repetitions
increases .
(e)
10,000 times produces empirical
The graph that represents __________
estimates closest to the actual probabilities.
(f)
Show your work for the calculation of the average (mean)
number of matches per repetition of the process.
mean = ________
4(
) + 3(0) + 2(
) + 1(
) + 0(
)
The long - run average value
achieved by a numerical random
process is called its expected value .
To calculate this expected value from
the (exact) probability distribution,
multiply each outcome by its
probability, and then add these up
over all of the possible outcomes.
(g)
[show your work]
4(.04)+3(0)+2(.25)+1(.33)+0(.38) = .99
æ 1ö
æ 1 ö æ 1ö
æ 3ö
4ç ÷ + 3(0) + 2ç ÷ + 1ç ÷ + 0ç ÷ = 1
è 24 ø
è 4 ø è 3ø
è 8ø
1
expected number of matches =________
[compare]
This should be reasonably close
to the simulated number.
Essential Question
What are the properties
of probability?
Activity 14-3
Weighted Coins
(pages 311 to 313)
The probabilities of landing heads for the six coins are:
coin A
¼
coin B
⅓
coin E
⅘
coin C
½
coin D
¾
coin F
99
100
(a) Take your guesses!
rep
1
2
3
4
5
relative
frequency
coin guess
(letter)
1st coin
H
H
T
H
H
2nd coin 3rd coin
H
H
H
H
H
T
T
H
T
T
4th coin
5th coin
6th coin
H
H
H
T
H
H
H
H
H
H
T
H
T
T
T
(a) Take your guesses!
rep
1
2
3
4
5
relative
frequency
coin guess
(letter)
1st coin
H
H
T
H
H
2nd coin 3rd coin
H
H
H
H
H
T
T
H
T
T
4th coin
5th coin
6th coin
H
H
H
T
H
H
H
H
H
H
T
H
T
T
T
0.8 1.0 0.2 0.8 1.0 0.2
NO
(b) Are you confident that all of your guesses are correct? ________
[explain]
The confidence of your guesses
is most likely hindered by the
duplicated frequencies.
(c) Again, take your guesses!
n = 10
1st coin
relative
frequency
0.70 0.90 0.20 0.80 1.00 0.20
coin guess
(letter)
2nd coin 3rd coin
4th coin
5th coin
6th coin
(d) And again, take your guesses!
n = 25
1st coin
2nd coin 3rd coin
4th coin
5th coin
6th coin
relative
frequency
0.56 0.88 0.28 0.88 1.00 0.20
coin guess
(letter)
n = 50
1st coin
relative
frequency
0.58 0.92 0.26 0.78 1.00 0.32
coin guess
(letter)
2nd coin 3rd coin
4th coin
5th coin
6th coin
(e)
After a total of 50 flips, are you reasonably
confident that your guesses are correct?
yes
/ no
________
[explain]
The confidence that your
guesses are correct should
increase as the number of flips
increase. The frequencies
should be getting closer to the
actual probabilities.
Here are the correct answers.
n = 50
1st coin
relative
frequency
0.58 0.92 0.26 0.78 1.00 0.32
coin guess
(letter)
C
2nd coin 3rd coin
E
coin C
½
A
4th coin
D
coin A
¼
coin E
⅘
5th coin
F
6th coin
B
coin F
99
100
coin D
¾
coin B
⅓
(f)
long - term
Probability describes the ________
behavior of random processes and not the
short - term behavior because many trials
_______
are needed before an accurate probability can
be deduced. For instance, this graph shows a
lot of variation in relative frequencies near
0 repetitions. However, as the six coins
_____
are flipped more often, the relative
stabilize (level off) and
frequencies _______________________
show us the long-term behavior we are
interested in.
Assignment
Activity 14-6: Random Cell Phones
(page 318 & 319)
Assignment
Activity 14-7: Equally Likely Events
(page 319)
Word Master Activity Worksheet
What are the properties of probability?
Essential Question
What are the properties
of probability?
Activity 14-4
Boy and Girl Births
(pages 314 to 316)
For simplicity we will assume that
the probability of a boy birth is 50%
independently from child to child.
(a)
Use Table I: Random Number Table on page
587. Pick a line, write down the first 4 digits,
then let even # = girl and odd # = boy.
Family 1
Child number
Random Digit
Gender
1
2
3
4
# girls
I am using line # 7.
Family 1
Child number
Random Digit
Gender
1
8
2
3
3
5
4
9
# girls
I am using line # 7.
Family 1
Child number
Random Digit
Gender
1
8
G
2
3
B
3
5
B
4
9
B
# girls
even # = girl and odd # = boy
I am using line # 7.
Family 1
Child number
Random Digit
Gender
1
8
G
2
3
B
3
5
B
4
9
B
# girls
1
even # = girl and odd # = boy
(b)
Repeat this for a total of 5 families, recording
the number of girls in each family.
even # = girl and odd # = boy
Family 1
Child number
Random Digit
Gender
Family #
# of girls
1
2
3
4
# girls
1
2
3
4
5
I am continuing on line # 7.
Family 1
Child number
Random Digit
Gender
Family #
# of girls
1
8
G
2
3
B
3
5
B
4
9
B
# girls
1
2
3
4
5
1
4487
1
I am continuing on line # 7.
Family 1
Child number
Random Digit
Gender
Family #
# of girls
1
8
G
2
3
B
3
5
B
4
9
B
# girls
1
2
3
4
5
1
3
2096
1
I am continuing on line # 7.
Family 1
Child number
Random Digit
Gender
Family #
# of girls
1
8
G
2
3
B
3
5
B
4
9
B
# girls
1
2
3
4
5
1
3
3
6323
1
I am continuing on line # 7.
Family 1
Child number
Random Digit
Gender
Family #
# of girls
1
8
G
2
3
B
3
5
B
4
9
B
# girls
1
2
3
4
5
1
3
3
2
1
9724
I am continuing on line # 7.
Family 1
Child number
Random Digit
Gender
Family #
# of girls
1
8
G
2
3
B
3
5
B
4
9
B
# girls
1
2
3
4
5
1
3
3
2
2
1
(c)
Combine your results with the rest of the class,
then divide by the total of simulated families to
get the empirical results.
Everyone will enter 5 tally marks!
Family #
# of girls
1
2
3
4
5
# of girls in family
0
1
2
3
4
number of
simulated
families
Empirical
probability
(c)
Combine your results with the rest of the class,
then divide by the total of simulated families to
get the empirical results.
Everyone will enter 5 tally marks!
Family #
# of girls
1
2
3
4
5
1
3
3
2
2
# of girls in family
0
1
|
2
3
4
number of
simulated
families
Empirical
probability
(c)
Combine your results with the rest of the class,
then divide by the total of simulated families to
get the empirical results.
Everyone will enter 5 tally marks!
Family #
# of girls
1
2
3
4
5
1
3
3
2
2
# of girls in family
0
1
|
2
||
3
4
number of
simulated
families
Empirical
probability
(c)
Combine your results with the rest of the class,
then divide by the total of simulated families to
get the empirical results.
Everyone will enter 5 tally marks!
Family #
# of girls
1
2
3
4
5
1
3
3
2
2
# of girls in family
0
1
|
2
||
3
||
4
number of
simulated
families
Empirical
probability
Enter your 5 tally marks!
# of girls in family
number of
simulated
families
Empirical
probability
0
1
2
3
|
||
||
4
In situations like this where
each trial has only two
possible outcomes and the
probabilities remain the same
on each trial independently
from trial to trial, exact
probabilities can be calculated
from
the binomial distribution .
Sample Space for a Family with 4 Children
0 girls
BBBB
4C 0
=1
1 girls
2 girls
3 girls
4 girls
Sample Space for a Family with 4 Children
0 girls
1 girls
BBBB
GBBB
BGBB
BBGB
BBBG
4C 1
=4
2 girls
3 girls
4 girls
Sample Space for a Family with 4 Children
0 girls
1 girls
2 girls
BBBB
GBBB
GGBB
BGBB
GBGB
BBGB
GBBG
BBBG
BGGB
BGBG
BBGG
4C 2
=6
3 girls
4 girls
Sample Space for a Family with 4 Children
0 girls
1 girls
2 girls
3 girls
BBBB
GBBB
GGBB GGGB
BGBB
GBGB GGBG
BBGB
GBBG GBGG
BBBG
BGGB BGGG
BGBG
BBGG
4C 3
=4
4 girls
Sample Space for a Family with 4 Children
0 girls
1 girls
2 girls
BBBB
GBBB
GGBB GGGB GGGG
BGBB
GBGB GGBG
BBGB
GBBG GBGG
BBBG
BGGB BGGG
n = 16
3 girls
4 girls
BGBG
BBGG
4C 4
=1
# of girls in family
0
Exact probabilities .0625
1
16
1
2
0 girls
1 girls
2 girls
BBBB
GBBB
GGBB GGGB GGGG
BGBB
GBGB GGBG
BBGB
GBBG GBGG
BBBG
BGGB BGGG
BGBG
BBGG
3 girls
3
4 girls
4
# of girls in family
0
1
Exact probabilities .0625 0.25
4
16
2 girls
2
3 girls
3
0 girls
1 girls
BBBB
GBBB GGBB GGGB GGGG
BGBB GBGB GGBG
BBGB GBBG GBGG
BBBG BGGB BGGG
BGBG
BBGG
4 girls
4
# of girls in family
0
1
2
Exact probabilities .0625 0.25 0.375
6
16
2 girls
3 girls
3
0 girls
1 girls
BBBB
GBBB GGBB GGGB GGGG
BGBB GBGB GGBG
BBGB GBBG GBGG
BBBG BGGB BGGG
BGBG
BBGG
4 girls
4
# of girls in family
0
1
2
3
Exact probabilities .0625 0.25 0.375 0.25
4
16
0 girls
1 girls
2 girls
BBBB
GBBB
GGBB GGGB GGGG
BGBB
GBGB GGBG
BBGB
GBBG GBGG
BBBG
BGGB BGGG
BGBG
BBGG
3 girls
4 girls
4
# of girls in family
0
1
2
3
4
Exact probabilities .0625 0.25 0.375 0.25 .0625
1
16
0 girls
1 girls
2 girls
BBBB
GBBB
GGBB GGGB GGGG
BGBB
GBGB GGBG
BBGB
GBBG GBGG
BBBG
BGGB BGGG
BGBG
BBGG
3 girls
4 girls
# of girls in family
0
1
2
3
4
Exact probabilities .0625 0.25 0.375 0.25 .0625
0 girls
1 girls
2 girls
BBBB
GBBB
GGBB GGGB GGGG
BGBB
GBGB GGBG
BBGB
GBBG GBGG
BBBG
BGGB BGGG
BGBG
BBGG
3 girls
4 girls
# of girls in family
0
1
2
3
4
Exact probabilities .0625 0.25 0.375 0.25 .0625
# of girls in family
0
1
2
3
4
number of
simulated
families
Empirical
probability
(d) [comment]
The class results should be fairly
close to the exact probabilities.
(e) According to the exact probabilities, is it more likely for a
family to have two children of the same gender or three
children of the same gender?
2 or 3
[explain]
The probability of having 2 children
of the same gender is 37.5%. There
are 2 ways to accomplish having 3
children of the same gender, …
1 girl and 3 boys or 3 girls and 1 boy.
Therefore, the probability of having 3
children of the same gender is 50%.
(f) Do you expect the likelihood of an exact 50/50 gender split to
increase or to decrease with larger families?
increase or decrease
[explain]
Answer this question as you
believe it should be answered.
Please, completely explain your
answer. I will not take points off if
you are wrong.
(g) Follow the directions to do the calculations on your
graphing calculator.
 CHOOSE A PARTNER TO RUN THE SIMULATION WITH
4 CHILDREN AND THE OTHER RUN THE
SIMULATION WITH 10 CHILDREN.
 YOU WILL NEED A SIGNIFICANT AMOUNT OF
MEMORY ON YOUR CALCULATOR TO DO RUN THIS
SIMULATION!!! PLEASE REMOVE ANY
UNNECESSARY LISTS BEFORE RUNNING THE THE
SIMULATION.
Press…
MATH → PRB ↓ 7:randBin(
Enter … 4 (or 10) , .5 , 500 )
then press ENTER
randBin(4,.5,500)
# of children
# of observations
4 or 10
to simulate the
# of families
probability
randBin(4,.5,500)
{3 1 3 2 2 2 1 …
Store into
a list by
using the
STO key.
randBin(4,.5,500)
{3 1 3 2 2 2 1 …
AnsL1
{3 1 3 2 2 2 1 …
Make a histogram and adjusting the
WINDOW to Xmin=0, Xmax=5 (or 11),
and Xscl=1, then press GRAPH.
The proportion of the 4-children families that
have 2 boys and 2 girls is …
___________.
Look at the bar
(exact = 37.5%)
for 2 children.
The proportion of the 10-children families that
have 5 boys and 5 girls is …
___________.
Look at the bar
(exact = 24.6%)
for 5 children.
confirmed or refuted
My expectation in (f) was __________________.
Assignment
Activity 14-8: Interpreting Probabilities
(page 319)
What are the properties of probability?
Essential Question
What are the properties
of probability?
Activity 14-5
Hospital Births
(pages 316 & 317)
(a) Proportion of days of days that hospital A observed an
equal count of girls is ________.
25%
90 ÷ 365 =
24.7%
Proportion of days
of days that hospital B observed an
13%
equal count of girls is ________.
46 ÷ 365 =
smaller12.6%
The ______________
hospital has more days with an
exact 50/50 split.
(b) Hospital ______
A has more days on which 60% or
more of the births are girls. (births ≥ 60%)
Hosp. A is 143 days and Hosp. B is 31 days
Is your prediction from the “Preliminaries”
yes / no
section supported?
________
(c) Hospital ______
B has more days on which between
41% and 59% of the births are girls.
(40% < births < 60%)
Hosp. A is 163 days and Hosp. B is 314 days
(d) While a __________
larger sample size
makes it less likely to get an exact
50/50 split in the observed counts, the
probability of getting a sample
proportion close to one-half increases
with a __________
larger sample.
This activity reveals that while a larger sample
size makes it less likely to get an exact 50/50 split
in the observed counts, the probability of
getting a sample proportion close to ½ increases
with a larger sample. Consequently, we are
less likely to obtain a sample proportion far
away from the long-term probability of ½.
Also note that a larger sample produces a
probability distribution that is quite
symmetric and mound shaped.
WRAP-UP
This topic has initiated your study of randomness by
introducing you to the concept of probability. You
have learned that probability is a long-run property
of events, and you have studied probability by
conducting simulations. You have carried out these
simulations using physical devices such as index
cards, using a table of random digits, and using
your calculator. You have also studied probability
more theoretically, through the notion of equally
likely events, sample space, and expected value.
Assignment
Activity 14-9: Racquet Spinning
(pages 319 & 320)
We will do part (d) to this activity as a class.
Everyone will get a chance to SPIN a racquet!
Tennis players will take the lead.
Your topic is due!
What are the properties of probability?