Download Statistics- Mathematical branch that deals with

Algebra 1
Lesson- Statistics Intro/Stat Graphs
Name:____________________________________
Date:_____________________________________
Objective:
To review the basic concepts of statistics and statistical graphs
Statistics-
Mathematical branch that deals w/: Planning experiments, Obtaining data, Organizing data,
Summarizing, Analyzing, Interpreting, and Drawing Conclusions based on data.
Census:
a collection of data from each member of a given population (each member of Black, Caucasian,
A/PI, and Hispanic is polled.)
Sample:
a relatively small grouping of individuals used to draw conclusions about the whole population.
Individuals: a member of a population (not necessarily a person).
Numerical Measurement
Parameter- a numerical value used to describe an entire population
Statistica numerical value used to describe a sample.
Types of Data
Quantitative data- data is represented numerically (counts or measurement)
Qualitative data- data can be separated into various categories with no numerical characteristics.
Two types of Quantitative data
Discrete- finite values (countable)
Ex:
number of wins in a baseball season
Number of students in a class
Count of eggs in a basket
Continuous- infinite (uncountable) values (varies based upon measuring device)
Ex:
time it takes to cross a street
Length of a person’s foot
Height, weight
Data Classification
There are four ways data can be classified.
1. Nominal level
2. Ordinal level
3. Interval level
4. Ratio level
1.
Nominal level:
consists of names, labels, or categories only; cannot assign numerical values.
2.
Ordinal level:
can be arranged in some order, but the differences between data values either
cannot be determined or are meaningless.
3.
Interval level:
can be arranged in some order and the differences between data values are
meaningful but there is no possible way for zero to be in the set.
4.
Ratio level:
can be arranged in some order and the differences between data values are
meaningful and zero can be in the set.
1
Examples:
Classify the following scenarios as being nominal, ordinal, interval, or ratio.
1.
Letter Grades
2.
Gender
3.
Weight (in carats) of diamonds
4.
Body temperatures.
OBSERVATIONAL STUDY V. EXPERIMENT
Observation: A non-intrusive approach to data collection.
Ex:
Observing the time it takes an individual’s passage through a maze
Experiment: Data collection that involves an imposition of a treatment. Helps determine cause-and-effect
relationships.
Ex:
Observing the time it takes an individual’s passage through a maze after you have placed a
hungry/angry lion in the maze.
Observational Study
Cross-sectional study- data are observed, measured, and collected at one point in time.
Retrospective study- data collected from past events (examination of records, etc.)
Longitudinal study- data are collected in the future from groups with common factors (cohorts)
Design of an Experiment
EU- Experimental Units (Individuals) on which the experiment is being conducted.
Subject- human EU
Treatment- a specific condition applied to an EU.
Explanatory variable- input variable
Response variable- output variable
Lurking variable- a variable that is unaccounted for in the design of your experiment.
Confounding- the presence of multiple lurking variables and/or you are unable to distinguish among the
effects of different factors.
Placebo- a dummy treatment.
2
Blinding- subject does not know if he/she is receiving a treatment or placebo.
Double-blinding- neither subject or person administering the test knows which is the placebo and which is the
treatment.
Blocking- a group of subjects that are similar.
Randomized Block design- 1. Form blocks from sample.
2. Randomly assign treatments to individuals within the blocks.
Exploratory Data Analysis: analysis in which there are no prior expectations. You are not looking for a
particular result.
Formal Statistical Inference: analysis in which you have specific questions that you would like answers to.
The purpose is to answer those questions to a reasonable degree of certainty.
Types of sampling
Anecdotal Evidence: evidence collected haphazardly with no real connection to the population. This data is
not representative of the population. An example of this is polling your friends for their opinion on smoking on
school grounds.
Available Data: Data produced in the past to answer someone else’s question but can also be used for your
study.
Random Samples
SRS- Simple random sample: Each member of the population is equally likely to be selected. Equivalent to
putting all the names of each person in the population in a hat and drawn a given number of names.
Stratified Random Sample- A Simple Random Sample is taken from each group in the population. This
insures equal representation among the different groups.
Systematic sampling- obtained by selecting every k th individual from the population.
Cluster sampling- obtained by selecting all individuals w/in a randomly selected collection or group of
individuals.
Convenience sampling- sample in which the individuals are easily obtained.
Matched Pairs- One sample is drawn and two conditions are imposed. Most usually, matched pairs refer to a
“before and after” situation.
Voluntary Response Sample (VRS)- Self selection. These samples are designed so that people with an ax to
grind or nothing better to do with their time can be polled. Ex: ESPN Web Poll, Radio show call-ins, etc.
3
Error and Bias
Sampling error- errors that result in the survey process. They are due to the non-response of individuals
selected, inaccurate responses, poorly worded questions, to bias in selection of surveyed individuals, etc.
Non-sampling error- errors that stem from incorrect collection of the data (human error)
Bias- sample is invalid because it is skewed in an inappropriate direction.
Process for conducting an experiment
1.
Identify the problem to be solved or claim to be tested (specifically the explanatory and response
variables)
2.
Determine the factors that affect the response variable.
3.
Determine the sample size (number of EUs)
4.
Determine the level of the response variable (nominal, ordinal, interval, ratio)
5.
Collect your sample using randomization and replication.
6.
Apply the selected treatment.
6.
Test the claim.
Example:
Design an experiment that will test the effectiveness of Vicodin (a pain relieving medication).
Your EU’s are the students in our class.
4
On your own:
1.
A researcher is attempting to determine if there is a correlation between upbringing and the committing
of sex crimes as an adult. Which of the following designs is the most appropriate for this observational
study?
(a)
Obtain a SRS of 4 year old children and follow their development up until the age of 18. Obtain police
records at ages 25, 35, 45, 55, 65…death to determine the number of sex crimes committed and analyze.
Obtain a stratified random sample of 4 year old children (strata determined by race) and follow their
development to the age the subject moves out on their own. Obtain police records at ages 25, 35, 45, 55,
65…death to determine the number of sex crimes committed and analyze.
Obtain a VRS of individuals who are age 50. Distribute a survey that, among other questions, polls
the individual as to whether or not they have committed sex crimes and analyze.
Obtain a SRS of 4 year old children and follow their development up until the age of 18. Observe each
individual as they live their daily lives, making certain to take special note of the committing of sex
crimes. Document and analyze.
(b)
(c)
(d)
2.
A study is conducted to determine if type of music listened to, has any affect on math test scores. Four
math classes are selected at random and given a test on basic math concepts. The students’ scores are
recorded and analyzed using comparative box-plots. A different type of music is played in each class
(rock, rap, country). Which of the following is most appropriate?
(a)
(b)
(c)
(d)
There is one explanatory variable and three response variables.
There are three levels of a single explanatory variable.
There are three explanatory variables.
There are three explanatory variables each with one treatment.
3. Which of the following are important in the design of experiments?
I.
Control of confounding variables
II.
Randomization in assigning subjects to different treatments
III.
Replication of the experiment using sufficient numbers of subjects
(A) I and II
(C) II and III
(B) I and III
(D) I, II, and III
(E) None of the above gives the complete set of true responses.
4. Which of the following are true about the design of matched-pair experiments?
I.
Each subject might receive both treatments.
II.
Each pair of subjects receives the identical treatment, and differences in their responses are
noted.
III.
Blocking is one form of matched-pair design.
(A) I only
(B) II only
(C) III only
(D) I and III (E) II and III
5
5.
A nutritionist believes that having each player take a vitamin pill before a game enhances the
performance of the football team. During the course of one season, each player takes a vitamin pill
before each game, and the team achieves a winning season for the first time in several years. Is this an
experiment or an observational study?
(A)
(B)
(C)
(D)
(E)
An experiment, but with no reasonable conclusion possible about cause and effect
An experiment, thus making cause and effect a reasonable conclusion
An observational study, because there was no use of a control group
An observational study, but a poorly designed one because randomization was not used
An observational study, thus allowing a reasonable conclusion of association but not of cause and
effect
6.
A town has one high school, which buses students from urban, suburban, and rural communities.
Which of the following sample is recommended in studying attitudes toward tracking of students in
honors, regular, and below-grade classes?
(A)
(B)
(C)
(D)
(E)
Convenience sample
Simple random sample (SRS)
Stratified sample
Systematic sample
Voluntary response sample
7.
A company has 1000 employees evenly distributed throughout five assembly plants. A sample of 30
employees is to be chosen as follows. Each of the five managers will be asked to place the 200 time
cards of their respective employees in a bag, shake them up and randomly draw out six names. The six
names from each plant will be put together to make up the sample. Will this method results a simple
random sample of the 1000 employees?
(A)
(B)
(C)
Yes, because every employee has the same chance of being selected.
Yes, because every plant is equally represented.
Yes, because this is an example of stratified sampling, which is a special case of simple random
sampling.
No, because the plants are chosen randomly.
No, because not every group of 30 employees has the same chance of being selected.
(D)
(E)
8.
In a study on the effect of music on worker productivity, employees were told that a different genre of
background music would played each day and the corresponding production outputs noted. Every
change in music resulted in an increase in production. This is an example of
(A)
(B)
(C)
(D)
(E)
the effect of a treatment unit.
the placebo effect.
the control group effect.
sampling error.
voluntary response bias.
6
9.
In one study on the effect that eating meat products has on weight level, an SRS of 500 subjects who
admitted to eating meat at least once a day had their weights compared with those of an independent
SRS of 500 people who claimed to be vegetarians. In a second study, an SRS of 500 subjects were
served at least one meat meal per day for 6 months, while an independent SRS of 500 others were
chosen to receive a strictly vegetarian diet for 6 months, with weights compared after 6 months.
(A)
(B)
(C)
(D)
(E)
The first study is a controlled experiment, while the second is an observational study.
The first study is an observational study, while the second is a controlled experiment.
Both studies are controlled experiments.
Both studies are observational studies.
Each study is part controlled experiment and part observational study.
10. Scenario: A student wishes to conduct a study to determine if there is a correlation between the number of
hours of homework assigned per week in math class and the grade earned in that class.
a.
Is this an observational study, or an experiment?
b.
What is the best (least-biased) method of sampling? What is the most practical method of
sampling that can be used in this particular case?
c.
Describe an effective method for conducting this study.
d.
What, if any, are the lurking variables?
e.
What do you expect the results of this experiment might be?
f.
Should randomization be used? Why or why not?
7
Stat Graphs
Given the following data, construct the following graphs:
1.
Histograms
a.
Frequency
b.
Cumulative Frequency
2.
Box plot
80
100
91
0
90
100
51
75
75
75
83
56
83
42
80
25
95
11
95
55
8
Algebra 1
Lesson- Measures of Central Tendency
Name:____________________________________
Date:_____________________________________
Objective:
To learn about measures of central tendency and outliers.
DO NOW:
Describe the process you would use to create a box plot.
Measures of Central Tendency
Mean, median, mode, range, and midrange.
Notation
  Sum
x  individual data value
n  number of values in the set
Mean: The average of all the values in the data set.
 x , where x (called x-bar) refers to the mean of the data set.
x
n
Ex: Find the mean of the data set:
15.1 17.5 18.3 14.1 3.2
6.0
Median: The middle value of the data set with numbers in ascending order.
Steps:
1. Put data in ascending (increasing) order.
2. Count to the middle value. If there is/are:
a. Only one value: that is the median.
b. Two values: take the average of the two numbers; that is the median.
Ex: Find the median of the data sets:
a.
5
2
3
6
b.
6
8
4
3
4
1
0
2
1
2
9
14
24
9
Mode: The most frequent value(s) in the data set (can also be more than one or none).
Ex: Find the mode of the data sets:
a.
5
7
7
3
1
2
b.
1
3
5
7
9
11
c.
1
1
3
3
7
8
9
Range: The difference between the highest and lowest values.
Ex: Find the range of the data set.
1
3
5
7
Outliers:
9
17
Any value that falls outside the range of ( LQ  1.5IQR ,UQ  1.5IQR ) is considered an outlier.
Ex: Determine if the following set consists of any outliers.
2
4
5
6
7
8
9
10
42
10
Algebra 1
Lesson- MCT with tables
Name:____________________________________
Date:_____________________________________
Objective:
To determine measures of central tendency with frequency tables.
Mean, Median, Mode, Range, and Midrange of Frequency Tables
Class (x)
1
2
3
4
5
Frequency (f)
10
3
1
8
5
Ex 1: Use the above frequency table to determine the Mean, Median, and Mode.
Ex 2: Find the mean, median, and mode of the following data set.
**Note: If the class interval is a range, use the class midpoint.
Class (x)
10-19
Frequency (f)
51
20-29
13
30-39
22
40-49
4
50-59
92
Weighted Mean: Used to determine a mean when different components have different weights.
x
 ( w  x)
w
Ex 3: A student had the following grade distribution in an Algebra class. What’s this student’s weighted mean?
Type of grade
Percent (w) Score (x)
Test Average
50
92
Quizzes/Independent work Average 20
95
Attendance/Participation
10
80
Homework
15
100
11
Algebra 1
Lesson- Percentiles
Name:____________________________________
Date:_____________________________________
Objective:
To learn to:
a) find the percentile rank of a given score
b) find a score given its percentile rank
DO NOW:
Find the mean, median, and mode of the data in the following table:
X 1 2 4 5
7 9
F 12 42 43 65 43 19
Percentiles
A percentile rank represents the percent of values below a certain score.
Given a data set, you can find the percentile rank of a value by counting the number of scores below it, dividing
by the total number of values in the data set (n) and converting that score into a percent rounded to the nearest
whole number.
Given a data set, you can find the value that is the i th percentile, where i is as whole percent value. Convert i to
a decimal and multiply by the number of values in the data set (n). The resulting value (position locator) can be
either a whole number or decimal value.
If the result is a whole number (m), use m and m + 1. These numbers are the locators for the
i th percentile. Find the m th and the (m  1) st term. These numbers make up the i th percentile.
OR
If the result is a decimal, round up, this is the locator for the i th percentile.
Ex: Using the table below:
a.
find the percentile rank of 94
b.
find the value that is the 35th percentile
c.
find the value that is the 32nd percentile
d.
find the percentile rank of 51
11
51
86
111
13
55
87
117
17
58
88
121
24
63
91
126
29
69
93
127
31
72
94
131
33
73
97
132
38
74
100
134
42
79
103
135
46
83
109
139
12
Algebra 1
Linear Correlation, Regression
Name:____________________________________
Date:_____________________________________
Correlation and Regression
Correlation- When one variable is related to another in some way
Scatterplot- A plot on an x-y plane, where (x, y) are paired data plotted as a single point
Types of plots
Perfect +
Strong +
Moderate +
Perfect -
Strong -
Moderate -
Exponential
Quadratic
None
13
Linear Correlation Coefficient (r)
measures the strength of the linear relationship between the given variables (AKA Pearson’s Product Moment
Correlation Coefficient)
r

n xy   x  y 

n  x 2   x 
2


n  y 2   y 
2
; where n is the number of ordered pairs
r 2 (in percent form) is the percent of variation in the y variable that can be explained by variation in the x
variable.
r
1
.75  r < 1
.50  r < .75
-.50 < r < .50
-.75 < r  -.50
-1 < r  -.75
-1
Type of correlation
Perfect +
Strong +
Moderate +
None
Moderate Strong Perfect -
Ex 1: Construct a scatterplot and compute the linear correlation coefficient between x and y.
x 1 2 3 5 9 11
y 3 5 7 10 16 20
14
Linear Regression
AKA- Least Squares Line, Line of Best Fit, Linear Regression Equation
This is the equation that best fits the data set given.
Calculator uses:
y = ax + b
where a = slope and b = y-intercept
Text uses:
yˆ  b0  b1 x where b1 = slope and b0 = y-intercept
slope  a  b1 
n xy   x  y 


n  x 2   x 
2
 y  x    x  xy

n x    x 
2
y  int ercept  b  bo
2
2
Ex 2: Using the data from Ex 1, write the equation of the line of best fit. Plot this line on your scatterplot as
verification.
Using the Graphing Calculator
1. Enter x values in L1 and y values in L2
2. Press 2 nd y 
3. Make sure all plots are off, if not, Press 4 (Plotsoff) Enter
4. Press 2 nd y 
5. Select 1
6. Turn this plot on by highlighting On and hitting Enter
7. Press the Down Arrow once and Press Enter
8. Verify that L1 and L2 are listed.
9. Press Zoom 9 (On your graph you should see a scatterplot)
10. 2 nd 0 Press DiagnosticsOn Press Enter twice
11. Press Stat, Right Arrow, 4 (LinReg(ax + b))
12. L1 , L2 , Y1 Enter (Correlation and Regression data is now presented)
13. Zoom 9 (Line should appear on the graph verifying that equation is correct)
Ex 3: Verify your answers to Ex 1 and Ex 2 using the graphing calculator.
15
Algebra 1
Lesson- Regression Analysis
Name:____________________________________
Date:_____________________________________
Other Regressions
There are other regressions that can be determined using the graphing calculator.
Quadratic
Exponential
Example:
A study was conducted to determine if there is a relationship between the number of cookies eaten on a
daily basis and an individual male’s weight. The data are given below.
# of cookies daily
Weight in lb
0
180
2
200
5
265
9
185
10
170
a. Construct a scatterplot (complete, with appropriate labels)
b. Visually determine the best fit model.
c. Determine, using your calculator, the best regression model. Why is the one you chose the best?
Justify your answer by showing all work.
d. Determine the equation of best fit.
e. What weight would you expect a person who eats 100 brownie bites per day to be?
16
Algebra 1
Review- Statistics Test
Name:____________________________________
Date:_____________________________________
Infant mortality rate (per 1,000 live births)
Country
1970
Country
2005
Iceland
13
Iceland
2
United Kingdom
18
Cyprus
4
Bulgaria
28
5
Cyprus
29
United
Kingdom
Croatia
Croatia
34
Bahrain
9
Qatar
45
Costa Rica
11
Lebanon
45
Bulgaria
12
Bahrain
55
13
Viet Nam
55
Argentina
59
Bosnia and
Herzegovina
Syrian Arab
Republic
Argentina
Bosnia and
Herzegovina
Costa Rica
60
Viet Nam
16
62
Qatar
18
Solomon Islands
70
Mexico
22
Jordan
77
Jordan
22
Mexico
79
Peru
23
Uzbekistan
83
24
Zimbabwe
86
Solomon
Islands
Lebanon
Syrian Arab
Republic
Kenya
90
Bangladesh
54
96
Uzbekistan
57
Refer to the table on the right to answer the questions
below.
1.
2.
6
3.
For the year 1970, determine the:
a.
Mean
b.
Median
c.
Mode
d.
Range
4.
Create a box plot (AKA Box Whisker Plot) and
list out the 5 number summary for 2005.
What country represents the 36th percentile for
1970?
What is the percentile rank of Lebanon for
2005?
What is the difference between discrete and
continuous data?
14
15
27
Congo
100
Tajikistan
59
Sudan
104
62
Tajikistan
108
Lao People's
Democratic
Republic
Sudan
Peru
119
Togo
78
Togo
128
Kenya
79
Lesotho
140
Congo
81
Lao People's
Democratic
Republic
Bangladesh
145
Zimbabwe
81
145
Burkina Faso
96
Côte d'Ivoire
158
Lesotho
102
Burkina Faso
166
Côte d'Ivoire
118
Niger
197
Niger
150
Sierra Leone
206
Sierra Leone
165
Create a frequency histogram for the year
2005.
Create a cumulative frequency histogram for
the year 1970.
5.
6.
7.
See reverse for #8-12
62
17
Producing Data
8.
A math teacher at a local high school is intrigued by the practice of “Irish step dancing.” He would like
to know if this is a method of dancing that would be difficult to master. To answer this question he
selects 100 students at random from the school population. Half of the students are instructed
extensively in the art of “Irish step dancing” by a student who is supposedly very good. The other half
are instructed by the math teacher after having watched a 1-hour video called “Irish Step Dancing:
Anyone Can Do It, Even Math Teachers!” A random selection of 5 individuals from each group, are
selected to perform and are rated by a group of 5 professional judges. Which of the following best
describes this situation?
(a) Observational study, no bias, no lurking variables.
(b) Experiment, explanatory variable: level of performance, response variable: type of training.
(c) Observational study, explanatory variable: type of training, response variable: level of performance.
(d) Experiment, explanatory variable: type of training, response variable: level of performance.
9.
Which among the following is most useful in establishing cause-and-effect relationships?
(A) A complete census
(B) A least squares regression line showing high correlation
(C) A simple random sample (SRS)
(D) A well-designed, well-conducted survey incorporating chance to ensure a representative sample
(E) A controlled experiment
10.
A consumer product agency tests miles per gallon for a sample of automobiles using each of four
different octanes of gasoline. Which of the following is true?
(A) There are four explanatory variables and one response variable.
(B) There is one explanatory variable with four levels of response.
(C) Mpg is the only explanatory variable, but there are 4 response variables corresponding to the
different octanes.
(D) There are four levels of a single explanatory variable.
(E) Each explanatory level has an associated level of response.
11.
A researcher is conducting a test to determine the effectiveness of a new experimental drug. This drug
is designed to eliminate headaches. Describe an effective method for conducting this study. Discuss in
particular randomization, lurking variables, treatments, and possible outcomes. Draw a diagram.
12.
Given the following set of data:
a.
Construct a scatterplot to describe the data set using the grid provided.
b.
Determine the appropriate regression model. Justify your answer.
c.
Graph your regression equation on the grid you used to construct your scatterplot.
d.
Predict the number of trees a person who is 42 years old would cut down. Is this number
reasonable? Why?
Age
Average # of trees
16
28
19
34
21
36
26
46
39
78
18
Algebra 1
Lesson- Intro to Probability
Objective:
Name:____________________________________
Date:_____________________________________
To review the fundamental aspects of probability.
Probability is a branch of math that deals with the chance of an event occurring.
Event- An event is a collection of outcomes of a procedure.
Sample Space- A sample space is a complete list of all possible outcomes in an
experiment or activity.
Notation
p  probabilit y
a, b, c denotes specific events
p (a )  probabilit y of event a occurring
p (~ a )  probabilit y of event a not occurring
There are a few ways to determine probability:
1.
Experimentation:
Ex:
Flip a coin 200 times to get an idea of what the probability of getting “Heads” when you flip a coin.
Law of Large Numbers:
As the number of trials in an experiment increase, the observed probability gets
closer and closer to the actual probability.
2.
Formulaic Approach: This is the most common method for determining probability.
# of ways a can occur
1 head 1
p(a) 
p (heads ) 
  0.5
total # of possible outcomes
2 sides 2
3.
Estimation/Pattern Recognition:
This is the least frequently used method (mathematically).
Ex:
Weather prediction based upon past experience (almanac). This is definitely not an exact science, but it is better
than nothing.
Basic Probability Rules
0  p  1 , therefore probabilities can be represented as fractions, decimals, or percents.
Complement Rule- p(a)  p(~ a)  1
Also- p(a)  1  p(~ a) and p(~ a)  1  p(a)
Ex:
Given a multiple choice question with 5 choices, what is the probability of getting the correct answer by
guessing? What is the probability of getting an incorrect answer by guessing? Verify the complement rule.
19
Rounding Rule- Round to 3 significant digits (usually means thousandths place but not always.
Ex:
.2380017  .238
.000072098  .0000721
To avoid confusion, perhaps it may be best to stay in fraction form unless you absolutely must use decimals.
Sample Space Information
Below you will find a listing of the most useful sample spaces for this course. You will need to have these
memorized.
1 Coin: {H, T}
2 Coins: {HH, HT, TH, TT}
3 Coins: {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}
1 Die: {1, 2, 3, 4, 5, 6}
2 Dice:
{1,1
2,1
3,1
4,1
5,1
6,1
1,2
2,2
3,2
4,2
5,2
6,2
Standard deck of 52 cards:
1,3
2,3
3,3
4,3
5,3
6,3
1,4
2,4
3,4
4,4
5,4
6,4
1,5
2,5
3,5
4,5
5,5
6,5
1,6
2,6
3,6
4,6
5,6
6,6}
4 suits: {hearts (r), diamonds (r), clubs (b), spades (b)}
13 cards per suit: {2,3,4,5,6,7,8,9,10,J,Q,K,A}
Prime Numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43…}
Sample Space Size
Flipping a coin:
One flip yields 2 outcomes, two flips yields 4 outcomes, three flips yields 8 outcomes. Is there a pattern?
Rolling a die:
One roll yields 6 outcomes, two rolls yields 36 outcomes…
Determine the number of values in the SS for three rolls of a die.
Is there a general formula we can use? What is it?
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Algebra 1
Lesson- Addition Rule
Name:____________________________________
Date:_____________________________________
Objective:
To review the use of the Addition Rule for single events.
DO NOW:
Determine the number of values in the sample space for 5 rolls of a fair die.
Addition Rule
This method is used to determine the probability in a single trial of either of two events occurring.
Ex 1:
A card is selected from a standard 52 card deck. Determine the probability of either a 7 or an Ace being
selected.
Ex 2:
A card is selected from a standard 52 card deck. Determine the probability of either a 7 or a red card being
selected.
What is the difference between the two above examples? Use Venn diagrams to support your answer.
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General Addition Rule
p(a or b)  p(a)  p(b)  p(a and b)
You can see that in some cases (like example 1) there is no overlap that needs to be subtracted. When that
occurs, we are dealing with Mutually Exclusive or Disjoint events.
Addition Rule for Mutually Exclusive Events
p(a or b)  p(a)  p(b)
Ex 3:
For each problem below, establish whether we are dealing with Mutually Exclusive events, then compute the
requested probabilities.
A card is drawn from a standard 52 card deck. Find:
p(7 or king )
a.
b.
p (7 or prime )
c.
d.
p(7 or not red )
p( prime or even)
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Algebra 1
Lesson- Multiplication Rule
Name:____________________________________
Date:_____________________________________
Objective:
To review the multiplication rule & complement rule for multiple events.
DO NOW:
Find the probability of selected a 7 or an Ace in one selection from a standard 52 card deck.
Multiplication Rule
This method is used in situations where 2 or more trials are being conducted.
In this case a tree diagram becomes a useful tool to get a look at the sample spaces for certain events.
Ex 1:
A coin is tossed then a die is rolled. Construct a tree diagram to illustrate p(toss a heads and roll a 5).
Determine the probability.
When all is said and done p(toss a heads and roll a 5)  p(toss a heads)  p(roll a 5)
In symbolic form: p(a and b)  p(a)  p(b)
This is called the Multiplication Rule for Independent Events.
Independent events occur when the probability of the second event occurring is not affected by the fact that the
first event has occurred.
Very frequently, however, we run into what are called Dependent Events. These usually occur when the
problem utilizes the words, “without replacement,” or words to that effect.
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Ex 2:
Two cards are drawn from a standard 52 card deck. The first card is not replaced before the second is drawn.
Find:
a. p(king and ace) in that order
b. p(king and ace) in either order
c. p(two cards with the same value)
d. p(two cards of the same suit )
When computing these probabilities, some adjustment must be made to the probability of “b” given that a
occurred. Therefore, we arrive at the following formula:
General Multiplication Rule
p(a and b)  p(a)  p(b | a); where p(b | a)  p(b " given" that a has already occurred )
When p (b | a )  p (b) the events are independent.
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Algebra 1
Lesson- Counting/Combinatorics
Name:____________________________________
Date:_____________________________________
Objective:
To learn the basics of combinatorics including:
 Factorials
 Permutations
 Combinations
DO NOW:
Three cards are selected from 2 standard 52 card decks (that are shuffled together) without
replacement. Find the probability that all three cards are prime numbers.
Counting/Combinatorics
Factorials, permutations, and combinations
Ex 1:
A group of three people must be arranged in a line. How many possible arrangements are there?
Factorials:
Ex 2:
5! = 5  4  3  2  1= 120
General Rule: n! n(n  1)( n  2)( n  3)...( n  n  1) but why would you ever want to remember that? Just
remember the above example.
Ex 3:
A line consisting of two people must be arranged from a group of three people. How many possible
arrangements are there?
Permutation- a limitation of a factorial.
5 P3  5  4  3  60
The first number indicates the initial value. The second value tells you how many values form your product.
This type of computation is done when you are arranging part of a group.
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General Rule: n Pr 
n!
(n  r )!
Ex 4: Five students are in a class. The teacher wants to know how many possible groupings of the three
students are possible. How many are there?
Ex 5: How many ways can you arrange the letters in the word “TRIAL” ?
Ex 6: How many ways can you arrange the letters in the word “KYRGYZSTAN” ?
Ex 7: Find the probability that of 5 selected people, no two share the same birthday.
Ex 8: Find the probability that of 25 selected people, no two share the same birthday.
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Algebra 1
Review- Probability Test
Name:____________________________________
Date:_____________________________________
Question Topic
#
1
Addition Rule
Example
Solution
A card is selected at random from a standard 52
card deck. Find the probability of selecting:
a) a jack or an ace
b) a jack or a red card
c) an orange card
A die is rolled and a spinner with 5 equal
regions (#1-5) is spun. Find the probability of:
a) rolling a 3 and spinning a 2
b) rolling an odd and spinning an even
c) rolling and spinning an odd
50 people are to be arranged in a line. What is
the total number of 3-person arrangements?
2
Multiplication
RuleIndependent
Events
3
Permutations
4
Factorials/
Permutations
5
Tree Diagram
6
Sample Space
Size
How many events are in the sample space for 5
flips of a coin?
7
Multiplication
RuleDependent
Events
Two cards are selected at random from a
standard 52 card deck, without replacement.
Find the probability of selecting:
a) a jack and then an ace
b) two red jacks
c) two spades
How many ways can you arrange the letters in
the word:
a) SPICY
b) CRUNCHY
Construct a tree diagram or list the sample space
for 1 roll of a die followed by a flip of a coin.
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