MDM 4U
Mathematics of Data Management
Exam Review and Outline
Name:
Unit 1: Permutations and Combinations
A. 4.1 – 4.3 Permutations
1. Know and be able to use for problems dealing with permutations:
a) Fundamental Counting Principle (Rule of Product): 𝒂 × 𝒃 × 𝒄 × …
b) Rule of Sum:
𝒂+𝒃+𝒄+⋯
c) Indirect Method: (total # possible outcomes) – (# outcomes not considered a success)
2. Be able to count using:
a) A list created from the given information
b) A tree diagram
c) Pigeon holes
3. Simplify and evaluate factorial expressions with and without a calculator.
RECALL:
n! = n (n-1) (n-2) (n-3) X ... X 2 X 1
0! = 1
4. Count the number of ordered arrangements or Permutations of a set of items.
5. Know, understand and be able to use the definition of “permutation”.
6. Notation and Formula: P(n, r) or
nPr
𝒏!
𝑷(𝒏, 𝒓) = (𝒏−𝒓)! , where r ≤ n and n, r ∈ 𝑁
7. Calculate the number of permutations when some objects are alike.
𝒏!
Number of permutations when there are like objects = 𝒂!𝒃!𝒄!…
B.
5.1 – 5.3 Combinations and Venn Diagrams
1. Set Notation, terminology, and definitions
n(A), A’, { }, 𝑨 ∩ 𝑩, 𝑨 ∪ 𝑩, 𝑨 ⊂ 𝑩, 𝚽, universal set
2. Be able to use and draw Venn Diagrams as a counting method.
3. Principle of Inclusion and Exclusion.
For two sets: 𝒏(𝑨 ∪ 𝑩) = 𝒏(𝑨) + 𝒏(𝑩) − 𝒏(𝑨 ∩ 𝑩)
For three sets:
𝒏(𝑨 ∪ 𝑩 ∪ 𝑪) = 𝒏(𝑨) + 𝒏(𝑩) + 𝒏(𝑪) − 𝒏(𝑨 ∩ 𝑩) − 𝒏(𝑨 ∩ 𝑪) − 𝒏(𝑩 ∩ 𝑪) + 𝒏(𝑨 ∩ 𝑩 ∩ 𝑪)
4. Know understand and be able to use the definition of “Combination”.
5. Use Combination Notation to count the number of combinations of n different objects taken r at a time.
𝒏
𝒏!
nCr = 𝑪(𝒏, 𝒓) = ( ) =
(𝒏−𝒓)!𝒓!
𝒓
6. The total number of subsets from a set with n elements is 𝟐𝒏 .
The number of ways to make a selection from a set of n elements is 2𝑛 − 1 since you must exclude {}.
7. The total number of combinations which can be made from p items of one kind, q items of another kind, r items
of another, and so on is:
(𝒑 + 𝟏)(𝒒 + 𝟏)(𝒓 + 𝟏) …
(𝒑 + 𝟏)(𝒒 + 𝟏)(𝒓 + 𝟏) … − 𝟏 if {} not counted.
or
Unit 2: Binomial Theorem and Pascal’s Triangle (4.4 – 4.5, 5.4)
1. Draw Pascal’s triangle for any number of given rows in numerical form and combination form.
2. State Pascal’s Formula and use to evaluate or simplify expressions.
3. State and use the following three properties of Pascal’s Triangles in words, combination notation and sigma
notation.
𝑛
𝑛
𝑛+1
a) Pascal’s Formula: ( ) + (
)=(
)
𝑟+1
𝑟
𝑟+1
𝑛
𝑛
𝑛
𝑛
𝑛
b) Sum of a Row:
( ) + ( ) + ( ) + ( ) + ⋯ + ( ) = 2𝑛
0
1
2
3
𝑛
𝑛
𝑛
𝑛
∑𝑟=0 ( ) = 2
𝑟
𝑛
𝑛
c) Symmetry Property: ( ) = (
)
𝑟
𝑛−𝑟
Be able to use the above properties to simplify given combination notation statements.
4. Solve question involving applying Pascal’s Triangle to count pathways. For example: movement of a checker,
moving from one point to another on a map, spelling a word using a array of letter.
5. State the Binomial Theorem in both expanded and Sigma Notation.
𝑛
(𝑎 + 𝑏)𝑛 = ∑ (𝑛𝑟)𝑎𝑟 𝑏 𝑛−𝑟
𝑟=0
𝑛
𝑛
𝑛
𝑛
(𝑎 + 𝑏)𝑛 = ( ) 𝑎𝑛 b0 + ( ) 𝑎𝑛−1 𝑏1 + ( ) 𝑎𝑛−2 𝑏2 + ⋯ + ( ) 𝑎0 𝑏 𝑛
0
1
2
𝑛
6. Identify and use the properties of a binomial expansion:
a) Number of terms in an expansion
b) The coefficient of a term in the expansion
c) The exponents of the variable in a term of the expansion
7. Create any given binomial expansion by using the Binomial Theorem.
Unit 3: Introduction to Probability (6.1 – 6.5)
1. Use and Understand Basic Probability Concepts
a) Probability Scale
b) Empirical vs Theoretical Probability
c) Complement of an Event
d) Subjective Probability
2. Understand and apply ODDS to solve problems.
a) Define odds in favour of an event occurring
b) Given the odds find the probability
c) Odds against an event occurring
3. Calculate Probabilities Using Counting
a) Calculate the probability of events using: permutations, combinations, tree diagrams
b) Calculate probability of an event using the indirect method (Birthday Paradox)
4. Calculate the probability of Independent Events
a) Define the meaning of compound and independents events
b) Probability of independent events
c) Calculate the probability of independent events using the product rule.
5. Calculate the probability of Dependent Events using Conditional Probability
a) Define dependent events and conditional probability
b) Calculating conditional probabilities
6. Calculate the probability Mutually Exclusive and non-mutually exclusive Events
a) define mutually exclusive events
b) calculate the probability that A or B will occur using the addition rule for mutually exclusive events.
c) calculate the probability that A or B will occur using Principle of Inclusion and Exclusion for nonmutually exclusive events.
Unit 4: Probability Distributions (7.1 -7.4)
1. Introduction to Probability Distributions:
a) Define and understand the meaning of: random variable, discrete random variable, continuous random
variable, expected value, probability distribution.
b) Tabulate and graph probability distributions.
c) Calculate the Expected Value of an experiment and interpret its meaning in the context of the problem.
d) Know and use the formulas for calculating probability and expected value.
2) Uniform Distribution
a) Define and recognize the characteristics of an experiment with a uniform distribution.
b) Know and use the formulas for calculating probabilities and expected value for a uniform distribution.
c) Use E(X) to determine whether a game is fair or determine what to charge to make a profit.
d) Tabulate and graph uniform probability distributions.
3) The Binomial Distribution
a) Know and recognize the 3 Characteristics required for an experiment to consist of Bernoulli Trials.
b) Define a Binomial Distribution, repeated trials, independent events, success or failure outcomes.
c) Know and use the formulas for calculating probabilities and expected value for a Binomial distribution.
d) Use E(X) to determine whether a game is fair or determine what to charge to make a profit.
e) Tabulate and graph Binomial probability distributions.
f) Know the relationship to Binomial Theorem:
4) Geometric Distributions (Waiting Times)
a) Define waiting times and geometric distribution.
b) Define the random variable X for geometric distribution.
c) Know and use the formulas for calculating probabilities and expected value for a Geometric
distribution.
c) Use E(X) to determine whether a game is fair or determine what to charge to make a profit.
d) Tabulate and graph Geometric probability distributions.
5) Hypergeometric Distributions
a) Define the conditions necessary for a Hypergeometric Distribution.
b) Know and use the formulas for calculating probabilities and expected value for a hypergeometric
experiment with “success/failure” outcomes:
c) Use E(X) to determine whether a game is fair or determine what to charge to make a profit.
d) Tabulate and graph Hypergeometric probability distributions.
e) Know and use the formulas for calculating probabilities for a hypergeometric distribution with more than 2
outcomes.
Unit 5: Statistics of One Variable (2.1 – 2.6)
1. Define, know the meaning of and be able to use the following terms:
Statistics, raw data, variable, continuous
variable, discrete variable, Frequency table, frequency diagram, histogram, bar graph, frequency polygon,
intervals, range, relative-frequency table, relative frequency diagram, cumulative frequency graph, categorical
data, circle graph, pictographs. (see pages 91 – 100 and glossary)
1. Data Analysis with Graphs
a) Continuous and discrete variables. Know how to create tables and graphs for both types of data.
b) Create Frequency or Relative Frequency tables and graphs.
c) Create Cumulative Frequency tables and graphs.
d) Understand how to group data into appropriate intervals.
e) Calculate the range of a set of data and use it to determine an appropriate interval length.
f) Data that can be put into categories can be presented using bar graphs, circle graphs (pie charts), and
pictographs.
2. Indices
a) Know examples of indices (see text and notes)
b) Create an appropriate index for a set of data
3. Sampling Techniques
a) Know and be able to identify the 7 different types of sampling techniques which can be used to collect data
about a population.
b) Suggest the appropriate type of same to use for a given scenario.
4. Bias in Surveys
a) Know and be able to identify the 4 different types of bias in surveys which can occur when collecting data
about a population.
b) Suggest ways to change a survey or data collection method to avoid bias caused by unintentional bias, or
leading and loaded questions in a given scenario.
5. Measures of Central Tendency
a) Know how to calculate the mean, median, and mode for a set of discrete data.
b) Know how to calculate the mean, median and mode for a set of grouped data.
c) Understand the difference between the mean of the population and the mean of a sample.
d) Use appropriate notation and formulas to calculate mean, median and mode.
e) Recognize outliers in a set of data and account for the effect they will have on the 3 measures of central
tendency.
f) Calculate a weighted mean for a set of data.
6. Measures of Spread
a) Be able to find the range of a set of data.
b) Be able to determine the 1st, 2nd, and 3rd quartiles of a set of data and use appropriate formulas and notation
to justify.
c) Determine the interquartile range of a set of data.
d) Create a Box and Whiskers Plot to illustrate the range, interquartile range and median of a set of data.
e) Calculate percentiles for a set of data and interpret their meaning.
f) Calculate the deviation of the data in a set.
g) Determine the standard deviation and variance of a set of data using appropriate formulas and notation for
discrete or grouped data. Use columns in a frequency table to aid in the calculations.
Unit 6: Statistics of Two Variables (3.1 – 3.5)
“The Joy of STATS” be familiar with the examples of use of statistics in different fields. I.e. What is google doing, how
is San Francisco using stats etc
1. Scatter Plots and Linear Correlation
a. Know the definitions of: Dependent (or responsive) variables (pg 160), Independent (or explanatory)
variables, Linear correlation, Perfect positive (or direct) linear correlation, Perfect Negative (or inverse) linear
correlation, a scatter plot, a line of best fit.
b. Be able to classify linear correlations as zero, weak, moderate, or strong.
c. The Correlation coefficient. Remember: you need the extra columns x2, y2 and xy.
d. Be able to use the TI-83 Calculator to find the line of best fit, r, and r2.
2. Linear Regression: Be able to define, Regression, Interpolation, Extrapolation, Least Squares Fit, Residual
3. See example 1 on page 173 for an example of how to determine the line of best fit. You will need the extra
columns x2 and xy for the line of best fit.
4. Non Linear Regression: Be able to define, Non-Linear Regression (pg 184), Coefficient of Determination, r2.
5. Know how to find r2 for :
a.
Exponential Regressions
b.
Power Regressions
c.
Quadratic Regressions
d.
e.
6. Cause and Effect: Be able to define and recognize
a. Cause and Effect Relationships
b. Common Cause Factors
c. Reverse Cause and Effect Relationships
d. Accidental Relationships
e. Presumed Relationships
Cubic Regressions
Linear Regressions
7. Also, be able to define: Extraneous variables, Experimental variables, Control group
Know the techniques used on pg 199 that recognize cause and effect relationships.
8. Critical Analysis
a. Be able to analyse the sample size and
techniques used to collect data.
b. Extraneous Variables and Sample Bias
c. Detecting a Hidden Variable
d. Know the critical questions to ask to assess a
statistical study.
Note: You will be required to calculate the correlation coefficient and line of best fit by hand using a table. You
will also be required to perform linear and non-linear regression using the TI-83. Review questions should be
completed using both methods.
Unit 7: Normal Distribution (8.1 – 8.4)
1. Continuous Probability Distributions
a) Identify a data set as continuous or discrete distribution.
b) Identify situations that give rise to common distributions. (Unimodal, Bimodal, Positive or negative
skewed, exponential, bimodal)
c) Identify a data set as having a uniform probability distribution, calculate the probability of all outcomes and
graph the probability distribution. Solve probability problems about the distribution.
2. Properties of Normal Distribution
a) Demonstrate an understanding of the properties of a normal distribution.
b) Understand the effect on the distribution of changing mean or standard deviation.
c) Be able to graph a sketch and lable a normal distribution given the mean and standard deviation of the
distribution.
d) Make probability statements about normal distributions.
e) Demonstrate an understanding of the properties of the standard normal distribution of Z-scores.
f) Be able to convert a normal distribution to z-scores and use the table to calculate probabilities.
3. Normal Sampling and Modelling
a) Understand the relationship between a sample and a population.
b) Make predictions about a population using the mean and standard deviation of the sample.
c) Be able to apply the normal model to a discrete situation, using the continuous approximation. Treat each
outcome as a unit interval.
4. Normal Approximation to the Binomial Distribution
a) Test whether a normal approximation is appropriate for a given experiment.
b) Calculate the mean and standard deviation of the data using n, p, and q.
c) Approximate discrete outcomes using a unit interval.
d) Solve probability problems related to binomial distributions.
Review Questions
Cumulative Review: Chapters 4 to 6
Page 364 # 1 – 10
Cumulative Review: Chapters 7 – 8
Page 476 # 1 - 5
Cumulative Review: Chapters 2 - 3
Page 218 # 11 - 18
Course Review:
Page 477 # 4, 6 - 11, 15 - 22, 24, 26 - 28
Note: 1) Solutions to the review will be posted on the internet.
2) Additional Review Questions will be provided after the Christmas Break