Download Example 1

Al Kamal American international School Azra
Math Department
Term III
Grade: 11 Booklets
1.
Student Name :______________________ Grade :__________
Using probability to make fair decisions
Aligned To Common Core Standard:
High School Statistics- HSS-MD.B.6
Types of Problems
There are three types of problems in this exercise:
1. Fair and unfair: This problem describes three selection processes, one of which is fair and the other two are not. The
student is asked to drag the selections correctly to determine which method is fair and why the other methods are not
fair.
2. Is it fair and probability: This problem describes a selection method, often using a chart of empirical data. The student is
asked to determine whether the choice process is fair, and also to find the probability of a particular event.
3. Select those that are fair: This problem describes several selection processes. The student is asked to select all of the
methods that are fair from the given list.
1. On the Fair and unfair problem something has to be moved, i.e., the answer is never given with everything in the
correct spot. Therefore if one of the answers is already lined up with the correct explanation, the other two must be
switched.
2. The problems sometimes require common knowledge of coins, cards, and dice.
Real-life Applications
1. Probability, along with decimals, percent’s, and fractions are used to determine the probability of a basketball player
making a shot.
2. Data and statistics appear in news reports and in the media every day.
3. Many of the problems in this exercise could be viewed as real-life applications.
4. Statistics can be seen more frequently than calculus in every day life.
Objectives
In this lesson, using proportionality and a basic understanding of probability, students
learn to make predictions and test conjectures about the results of experiments and
simulations. Students will:




identify situations in which prediction is useful.
investigate how elementary probabilities, experimental or theoretical, aid in
predicting future outcomes related to real-world issues.
use experimental probability to make predictions.
understand that what happens in a predictable pattern over a long period of time
can be used for decision-making purposes.
(1)

discover that the greater the number of trials in a random experiment, the closer
the experimental probability will be to the theoretical probability (the law of large
numbers).
Vocabulary







Compound Event: Two or more simple events.
Equally Likely: Two or more possible outcomes of a given situation that have the
same probability.
Outcome: One of the possible events in a probability situation.
Probability: A number between 0 and 1 used to quantify likelihood for processes
that have uncertain outcomes
Random Sample: A sample in which every individual or element in the population
has an equal chance of being selected. A random sample is representative of the
entire population.
Sample Space: The set of possible outcomes of an experiment; the domain of
values of a random variable.
Simple Event: One outcome or a collection of outcomes.
Example:
Two teams decided to play cricket. They want to decide who bats first. Robert and David are the
team captains. So, they have two suggestions to decide that who bats first. Decide whether the
suggestions are fair ways to make the decision.
The suggestions are given below:Robert: We can flip a coin. If it lands on heads then my team bats first. If it lands on tails then
David’s team bats first.
David: We can roll a single die. If it lands on either 1, 2, or 3 my team bats first. If the roll is a 4, 5, or
6 then Robert’s team bats first.
Explanation: Robert’s suggestion is to flip a coin and if it lands on head then his team bats first and
if it lands on tail then David’s team bats first. This is a fair method because each player has equal
chance.
If we select the David’s suggestion, roll a dice. If it lands on 1, 2, or 3 then his team bats first and if it
lands on 4, 5, or 6 then Robert’s team bats first. This is also a fair method as well. Everyone has an
equally fair shot to win.
(2)
Use Probability to make fair decisions Determine if the method stated is fair or unfair way to decide
the outcome.
Explain.
1) Everyone pays $1 for a raffle ticket. Copies of each ticket are placed in a box and someone who
did not buy a raffle ticket reaches in and chooses one at random. That ticket is the winner.
2) Two people find a $10 bill on the ground. Instead of splitting the money, they decide to shoot
basketball free throws to see who gets the money. The first person to make 3 baskets wins the
money.
3) Fred and Sam roll a die. If the number is prime Sam wins. If the number is composite Fred wins. If
a 1 is rolled they roll again. For an outcome to be settled fairly each person must have an equal
probability to earn the prize.
(3)
Normal Distribution
Aligned to Common Core standard S-ID
Measures of Central Tendency
Introduction
A measure of central tendency is a single value that attempts to describe a set of data by
identifying the central position within that set of data. As such, measures of central tendency are
sometimes called measures of central location. They are also classed as summary statistics. The
mean (often called the average) is most likely the measure of central tendency that you are most
familiar with, but there are others, such as the median and the mode.
The mean, median and mode are all valid measures of central tendency, but under different
conditions, some measures of central tendency become more appropriate to use than others. In
the following sections, we will look at the mean, mode and median, and learn how to calculate
them and under what conditions they are most appropriate to be used.
Mean (Arithmetic)
The mean (or average) is the most popular and well known measure of central tendency. x 1, x2, ..., xn, the
sample mean, usually denoted by is:
the population mean and not the sample mean, we use the Greek lower case letter "mu", denoted as µ:
. For example, consider the wages of staff at a factory below:
Staff
Salary
1
15k
2
18k
3
16k
4
14k
5
15k
6
15k
7
12k
8
17k
The mean salary for these ten staff is $30.7k
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9
90k
10
95k
As most workers have salaries in the $12k to 18k range.
The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better
measure of central tendency.
As we will find out later, taking the median would be a better measure of central tendency in this situation.
Another time when we usually prefer the median over the mean (or mode) is when our data is skewed (i.e.,
the frequency distribution for our data is skewed). If we consider the normal distribution - as this is the most
frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are
identical. Moreover, they all represent the most typical value in the data set. However, as the data becomes
skewed the mean loses its ability to provide the best central location for the data because the skewed data is
dragging it away from the typical value. However, the median best retains this position and is not as strongly
influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this
guide.
Median
The median is the middle score for a set of data that has been arranged in order of magnitude.
In order to calculate the median, suppose we have the data below:
65
55
89
56
35
14
56
55
87
45
92
We first need to rearrange that data into order of magnitude (smallest first):
14
35
45
55
55
56
56
65
87
89
92
Our median mark is the middle mark - in this case, 56.
This works when you have an odd number of scores,
When you have an even number of scores
example below:
65
55
89
56
35
14
56
55
87
45
We again rearrange that data into order of magnitude (smallest first):
14
35
45
55
55
56
56
65
87
89
take the 5th and 6th score in our data set and average them to get a median of 55.5.
(5)
Mode
The mode is the most frequent score in our data set. On a histogram it represents the highest bar
in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most
popular option. An example of a mode is presented below:
We are now stuck as to which mode best describes the central tendency of the data.
Skewed Distributions and the Mean and Median
We often test whether our data is normally distributed because this is a common assumption
underlying many statistical tests.
An example of a normally distributed set of data is presented below:
Data can be "distributed" (spread out) in different ways.
It can be spread out
more on the left
Or more on the right
Or it can be all jumbled up
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Data can be "distributed" (spread out) in different ways.
It can be spread out
more on the left
Or more on the right
Or it can be all jumbled up
But there are many cases where the data tends to be around a central value with no bias left or
right, and it gets close to a "Normal Distribution" like this:
A Normal Distribution
The "Bell Curve" is a Normal Distribution.
And the yellow histogram shows some data that
follows it closely, but not perfectly (which is usual).
It is often called a "Bell Curve"
because it looks like a bell.
The Normal Distribution has:

mean = median = mode

symmetry about the center

50% of values less than the mean
and 50% greater than the mean
(7)
Standard Deviations
Standard Deviation Formulas
The Standard Deviation is a measure of how spread out numbers are.
You might like to read this simpler page on Standard Deviation first.
But here we explain the formulas.
The symbol for Standard Deviation is σ (the Greek letter sigma).
This is the formula for Standard Deviation:
Say what? Please explain!
OK. Let us explain it step by step.
Say we have a bunch of numbers like 9, 2, 5, 4, 12, 7, 8, 11.
To calculate the standard deviation of those numbers:

1. Work out the Mean (the simple average of the numbers)

2. Then for each number: subtract the Mean and square the result

3. Then work out the mean of those squared differences.

4. Take the square root of that and we are done!
The formula actually says all of that, and I will show you how.
The Formula Explained
First, let us have some example values to work on:
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Example: Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
μ=7
Step 2.
So it says "for each value, subtract the mean and square the result", like this
Example (continued):
(9 - 7)2 = (2)2 = 4
(2 - 7)2 = (-5)2 = 25
... etc ...
And we get these results:
4, 25, 4, 9, 25, 0, 1, 16, 4, 16, 0, 9, 25, 4, 9, 9, 4, 1, 4, 9
Step 3. Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by how many.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use "Sigma":
The handy Sigma Notation says to sum up as many terms as we want:
(9)
Σ
=178
Which means: Sum all values from (x1-7)2 to (xN-7)2
Mean of squared differences = (1/20) × 178 = 8.9
(Note: this value is called the "Variance")
σ = √(8.9) = 2.983...
Sample Standard Deviation
But wait, there is more ...
... sometimes our data is only a sample of the whole population.
Example: Sam has 20 rose bushes, but only counted the flowers on 6 of them!
The "population" is all 20 rose bushes,
and the "sample" is the 6 bushes that Sam counted the flowers of.
Let us say Sam's flower counts are:
9, 2, 5, 4, 12, 7
(10)
We can still estimate the Standard Deviation.
But when we use the sample as an estimate of the whole population, the Standard Deviation
formula changes to this:
The formula for Sample Standard Deviation:
The important change is "N-1" instead of "N" (which is called "Bessel's correction").
The symbols also change to reflect that we are working on a sample instead of the whole
population:
 The mean is now
x (for sample mean) instead of μ (the population mean),
 And the answer is s (for Sample Standard Deviation) instead of σ.
But that does not affect the calculations. Only N-1 instead of N changes the calculations.
OK, let us now calculate the Sample Standard Deviation:
Step 1. Work out the mean
Example 2: Using sampled values 9, 2, 5, 4, 12, 7
The mean is (9+2+5+4+12+7) / 6 = 39/6 = 6.5
So:
x = 6.5
Step 2. Then for each number: subtract the Mean and square the result
Example 2 (continued):
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(9 - 6.5)2 = (2.5)2 = 6.25
(2 - 6.5)2 = (-4.5)2 = 20.25
(5 - 6.5)2 = (-1.5)2 = 2.25
(4 - 6.5)2 = (-2.5)2 = 6.25
(12 - 6.5)2 = (5.5)2 = 30.25
(7 - 6.5)2 = (0.5)2 = 0.25
Step 3. Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by how many.
But hang on ... we are calculating the Sample Standard Deviation, so instead of dividing by how
many (N), we will divide by N-1
Example 2 (continued):
Sum = 6.25 + 20.25 + 2.25 + 6.25 + 30.25 + 0.25 = 65.5
Divide by N-1: (1/5) × 65.5 = 13.1
(This value is called the "Sample Variance")
Step 4. Take the square root of that:
Example 2 (concluded):
s = √(13.1) = 3.619...
DONE!
(12)
Comparing
When we used the whole population we got: Mean = 7, Standard Deviation = 2.983...
When we used the sample we got: Sample Mean = 6.5, Sample Standard Deviation = 3.619...
Our Sample Mean was wrong by 7%, and our Sample Standard Deviation was wrong by 21%.
Why Take a Sample?
Mostly because it is easier and cheaper.
Imagine you want to know what the whole country thinks ... you can't ask millions of people, so
instead you ask maybe 1,000 people.
There is a nice quote (possibly by Samuel Johnson):
"You don't have to eat the whole ox to know that the meat is tough."
This is the essential idea of sampling. To find out information about the population (such as mean
and standard deviation), we do not need to look at all members of the population; we only need a
sample.
But when we take a sample, we lose some accuracy.
Summary
The Population Standard Deviation:
The Sample Standard Deviation:
(13)
Example
You and your friends have just measured the heights of your dogs (in millimeters):
The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.
Find out the Mean, the Variance, and the Standard Deviation.
Your first step is to find the Mean:
Answer:
Mean = 600 + 470 + 170 + 430 + 3005 = 19705 = 394
so the mean (average) height is 394 mm. Let's plot this on the chart:
Now we calculate each dog's difference from the Mean:
To calculate the Variance, take each difference, square it, and then average the result:
(14)
So the Variance is 21,704
And the Standard Deviation is just the square root of Variance, so:
Standard Deviation
σ = √21,704
= 147.32...
= 147 (to the nearest mm)
And the good thing about the Standard Deviation is that it is useful. Now we can show which heights
are within one Standard Deviation (147mm) of the Mean:
So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is
extra large or extra small.
Rottweilers are tall dogs. And Dachshunds are a bit short ... but don't tell them!
Now try the Standard Deviation Calculator .
(15)
But ... there is a small change with Sample Data
Our example has been for a Population (the 5 dogs are the only dogs we are interested in).
But if the data is a Sample (a selection taken from a bigger Population), then the calculation
changes!
When you have "N" data values that are:

The Population: divide by N when calculating Variance (like we did)

A Sample: divide by N-1 when calculating Variance
All other calculations stay the same, including how we calculated the mean.
Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of
5 like this:
Sample Variance = 108,520 / 4 = 27,130
Sample Standard Deviation = √27,130 = 164 (to the nearest mm)
Think of it as a "correction" when your data is only a sample.
Formulas
Here are the two formulas, explained at Standard Deviation Formulas if you want to know more:
The "Population Standard Deviation":
The "Sample Standard Deviation":
Looks complicated, but the important change is to
divide by N-1 (instead of N) when calculating a Sample Variance.
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Example: Your score in a recent test was 0.5 standard deviations above the average, how many
people scored lower than you did?

Between 0 and 0.5 is 19.1%

Less than 0 is 50% (left half of the curve)
So the total less than you is:
50% + 19.1% = 69.1%
In theory 69.1% scored less than you did (but with real data the percentage may be different)
A Practical Example: Your Company packages sugar in 1 kg
bags.
When you weigh a sample of bags you get these results:

1007g, 1032g, 1002g, 983g, 1004g, ... (a hundred measurements)

Mean = 1010g

Standard Deviation = 20g
Question 1
)
What is the population standard deviation for the numbers: 75, 83, 96, 100, 121 and 125?
(17)
Question 2
Ten friends scored the following marks in their end-of-year math exam:
23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%
What was the standard deviation of their marks?
Question 3
A booklet has 12 pages with the following numbers of words:
271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314
What is the standard deviation number of words per page?
Question 4
The standard deviation of the numbers 3, 8, 12, 17 and 25 is 7.56 correct to 2 decimal places.
Use the standard deviation calculator (link below) to see what happens if each of the five numbers is
increased by 2.
(18)
Vectors in the Plane
Aligned to Common Core standard N-CN
A vect or i s a l i ne se gm ent runni n g f rom poi nt A (t ai l ) t o poi nt B (he ad).
Each ve ct or has a m agni t ude (al so re fer r ed t o as l en gt h) and a di rect i on.
Direction of a Vector
Thi s i s t he di rect i on of t he l i ne whi ch c ont ai ns t he vect or o r an y l i ne whi ch i s
paral l el t o i t .
Magnitude of a Vector
The magnitude of the vector
denoted by
is the length of the line segment
.
The magnitude of a vector is always a positive number or zero.
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. It is
The magnitude of a vector can be calculated if the coordinates of the
endpoints are known:
Examples
C al cul at e t he m a gni t ude of t he fol l owi n g ve ct ors:
C al cul at e t he val ue of k knowi ng t h e m a gni t ude of t he v ect o r
= (k, 3) i s 5.
Position Vector
The vect or
t hat j oi ns t he coordi nat e’s o ri gi n, O , wi t h a poi nt , P, i s t he posi t i on vect or of
t he poi nt P .
(20)
Components or Coordinates of a Vector
If t he coordi n at es of A and B are:
Examples
Fi nd t he com ponent s of t he vect or
The vect or
:
has t he com ponent s (5, −2). Fi nd t he coordi nat e s of A i f t he t erm i na l
poi nt i s known as B ( 12, −3).
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C al cul at e t he coo rdi nat es of P oi nt D so t hat t he quadri l at er a l of P oi nt s: A(−1, −2),
B(4, −1 ), C (5, 2) an d D form a pa ral l el o gr am .
(22)
Matrices
A Matrix is an array of numbers:
A Matrix
(This one has 2 Rows and 3 Columns)
We talk about one matrix, or several matrices.
There are many things we can do with them ...
Adding
To add two matrices: add the numbers in the matching positions:
These are the calculations:
3+4=7 8+0=8
4+1=5 6-9=-3
The two matrices must be the same size, i.e. the rows must match in size, and the columns must
match in size.
Example: a matrix with 3 rows and 5 columns can be added to another matrix of 3 rows and 5
columns.
But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size)
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Negative
The negative of a matrix is also simple:
These are the calculations:
-(2)=-2 -(-4)=+4
-(7)=-7 -(10)=-10
Subtracting
To subtract two matrices: subtract the numbers in the matching positions:
These are the calculations:
3-4=-1
8-0=8
4-1=3 6-(-9)=15
Note: subtracting is actually defined as the addition of a negative matrix: A + (-B)
Multiply by a Constant
We can multiply a matrix by some value:
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These are the calculations:
2×4=8
2×0=0
2×1=2 2×-9=-18
We call the constant a scalar, so officially this is called
"scalar multiplication".
Multiplying by another Matrix
To multiply two matrices together is a bit more difficult ... read Multiplying Matrices to learn
how.
Dividing
And what about division? Well we don't actually divide matrices, we do it this way:
A/B = A × (1/B) = A × B-1
where B-1 means the "inverse" of B.
So we don't divide, instead we multiply by an inverse.
...
Notation
A matrix is usually shown by a capital letter (such as A, or B)
Each entry (or "element") is shown by a lower case letter with a "subscript" of row,column:
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Example:
B=
Here are some sample entries:
b1,1 = 6
b1,3 = 24
b2,3 = 8
(the entry at row 1, column 1 is 6)
(the entry at row 1, column 3 is 24)
(the entry at row 2, column 3 is 8)
Question 1
A
C
B
D
Question 2
A
C
B
D
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Question 3
A
C
B
D
Question 4
A
C
B
D
Question 5
A
C
B
D
(27)
Multiplication of Matrices
Example 1
If we multiply a 2×3 matrix with a 3×1 matrix, the product matrix is 2×1
the product of matrix A and Matrix B below
defined or undefined?
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TRIGONOMETRY
THE UNIT CIRCLE Aligned to Common Core standard F-TF
Te definitions
The signs in each quadrant
Quadrantal angles
The unit circle
Let a radius of length r sweep out an angle θ in standard position,
and let its endpoint have coördinates (x, y). The question is:
How shall we now define the six trigonometric functions of θ?
We will take our cue from the first quadrant. In that quadrant,
According to the Pythagorean theorem,
Example 1. A straight line inserted at the origin terminates at the point (3, 2) as it sweeps out an
angle θ in standard position. Evaluate all six functions of θ.
Answer. x = 3, y = 2. Therefore, according to the definitions:
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sin θ =
y
2
=
r
csc θ =
r
=
y
2
cos θ =
x
3
=
r
sec θ =
r
=
x
3
tan θ =
y
2
=
x 3
cot θ =
x 3
=
y
2
Problem 1. A straight line from the origin sweeps out an angle θ, and it terminates at the point
(3, −4). Evaluate the six functions of θ.
x = 3, y = −4. Therefore,
sin θ = − 45
cos θ =
3
5
tan θ = − 43
csc θ = − 5
4
sec θ =
5
3
cot θ = − 3
4
Problem 2. The signs in each quadrant.
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Problem 5. Evaluate the following. No tables
a) cos 0°
cos 0° = 1. cos θ is equal to the x-coördinate.
b) cos 90° = 0
c) cos 180° = −1
d) cos 270° = 0
e) tan 0°
tan 0° = 0. tan θ is equal to y/x = 0/1 = 0.
f) tan 90° 1/0 does not exist.
g) tan 180° = 0
h) tan 270° does not exist.
Problem 6. Evaluate the following -- if it exists. No tables.
a) cos π
2
b) sin π = 1
2
c) sin π = 0
e) cot 0 = x/y. Does not exist.
g) tan π Does not exist.
2
d) cos π = −1
f) cot π = 0
2
h) sec 0 = 1/x = 1
j) sin 2π = 0
k) sin 3π = 0
n) cos 2π = 1
o) cos 3π = −1
l) sin 4π = 0
p) cos 4π = 1
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i) csc (− π ) = −1
2
m) sin (−π) = 0
q) cos 5π = −1
Law of Sines and Cosines
How to determine which formula to use
Law of Sine
The goal of this page is to help students better understand when to use the law of sines and when to use
the Law of Cosines
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Practice Problems
Problem 1
Look at each triangle below and, based on the given information, decide whether you could use the Law of Sines,
the Law of Cosines (or neither)
SHOW ANSWER
Problem 2
Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide
that, try to set up the equation(Do not solve--just substitute into the proper formula)
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SHOW ANSWER
Problem 3
Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide
that, try to set up the equation(Do not solve--just substitute into the proper formula)
SHOW ANSWER
Problem 4
Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide
that, try to set up the equation(Do not solve--just substitute into the proper formula)
SHOW ANSWER
.
Application of the Standard Law of Sines –
In ∆ABC, side a = 5, Angle C = 48°Angle A = 70°
Find side c to the nearest tenth of an integer.
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Area of a triangle
Area =
Example
Find the area of triangle ABC
Area =
Substituting in:
Area =
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Quadratic Equation and Quadratic Function
Standard Form:
Solving the quadratic equation
A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be
distinct, and they may or may not be real.
Factoring by inspection:
For the quadratic function y = x2 − x − 2, the points where the graph crosses the x-axis, x = −1and x = 2,
are the solutions of the quadratic equation x2 − x − 2 = 0.
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Completing the square:
Solving
2x2 + 4x − 4 = 0
X² + 2x −2 = 0
X² + 2x =
X² + 2x +1 = 2 +1
(X+1)² =3 …………………………………
X = -1 ±√3
Quadratic formula:
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
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Discriminant[edit]
In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation,
and is often represented using an upper case D or an upper case Greek delta:[9]
D= b² - 4ac
Quadratic Function:
f(x) = ax²+bx +c
The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis,
Graph of the function:
Vertex:
x=
The vertex, (h, k), of the parabola in standard form is
(
−𝒃
𝟐𝒂
−𝒃
𝟐𝒂
,c-
−𝒃²
𝟒𝒂
𝟐𝒂
,c-
Maximum and minimum points:
(
−𝒃
)
(38)
−𝒃²
𝟒𝒂
)
Solve the following equations by any way
1- x2 + 4x - 72 = 5x
2- x2 - 10x - 11 = 0
3- x2 + 4x - 21 = 0
4- x2 - 4x - 32 = 0
5- x2 + 71 = -18x - 9
Graph each equation.
y = x2 - 8x + 10
y = 3x2 - 6x + 1
(39)
y = -x2 + 6x - 8
Question 1) Solve: x2 + 5x + 6 = 0
Question 4) t2 + 2t - 19 = 5
Question 7) 2x2 + 6x + 4 = 0
Question 16) 4n2 + 12n + 9 = 0
Question 20) 3x3 + 21x2 + 36x = 0
(40)
End