Download Analytic Geometry Condensed Study Guide

Analytic Geometry Condensed Study Guide
Unit 1
A dilation is a transformation
that makes a figure larger or
smaller than the original figure
based on a ratio given by a scale
factor. The dilation is an
enlargement when the scale
factor is greater than 1, a
reduction when it is between 0
and 1.
__________________________
multiply the coordinates by the
scale factor.
To determine the center of
dilation, connect each of the
image points to the preimage
points and note the intersection
point.
In the figure above, <1 and <5
are corresponding, <3 and <5
are alternate interior, and <1 and
<3 are vertical. All of these
relationships yield congruent
angles. The bold arrows in the
lines indicate the lines are
parallel.
Here, the scale factor is 2. If the
center of dilation is the origin,
you can just multiply all the
coordinates by the scale factor.
ABC is the original picture or
preimage, A’B’C’ is the new
figure or image.
Notice the diagram and the
proof that follows:
When a figure is dilated, the
image and preimage are
similar(same shape). Similar
figures have congruent angles
and proportional sides.
__________________________
If a segment of a figure dilated
does not pass through the center
of dilation, the segment will
shift to a parallel segment(see
̅̅̅̅
𝐴𝐵 and ̅̅̅̅̅̅
𝐴′𝐵′ above). If the line
passes through the center of
dilation, it will not move, as
seen in the diagram below(note
̅̅̅̅̅̅̅:
̅̅̅̅ is on 𝐴′𝐶′)
𝐴𝐶
Two triangles can be proven
similar if you can find two pair
of corresponding angles
congruent. This is called
AA(angle-angle).
Given:
Prove:
Note the triangles are the same
shape but a different size.
To prove that two triangles are
similar(or congruent), a twocolumn proof is often used.
Note that if the center of dilation
is not (0,0), you do not simply
If parallel lines are present
within triangles, look for
alternate interior or
corresponding angles. Look
also for vertical angles(opposite
in an x shape) and shared sides.
The missing reason would be
“Corresponding Angles are
Congruent.”
Figures are congruent if they
have the same size and shape.
Congruent figures have
congruent sides and angles.
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Analytic Geometry Condensed Study Guide
A rigid motion is a
transformation of points in
space consisting of a
sequence of one or
more translations,
reflections, and/or
rotations (in any order).
This transformation leaves
the size and shape of the
original figure unchanged.
Translation,Reflection,Rot
ation → Congruent and
Similar
Dilation→Similar
If figures are congruent, they are
also similar. Any of the six
corresponding parts(three angles
and three sides) of two
congruent triangles are
congruent. This is often called
CPCTC (corresponding parts of
congruent triangles are
congruent.)
A problem on a grid can be
shown congruent by counting
blocks.
The following patterns are
sufficient for showing two
triangles congruent:
Triangle Midsegment
Theorem: If a segment
joins the midpoints of two
sides of
a triangle, then the
segment is parallel to the
third side and half its
length.
Above, ̅̅̅̅
𝐷𝐸 is parallel to ̅̅̅̅
𝐵𝐶 and
is half its length.
The three medians of a triangle
meet at a point called the
centroid.
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
The two triangles above can be
shown congruent by SAS by
simply counting off the
horizontal and vertical lengths.
The three angles of a triangle
add to 180 degrees.
ASA (Angle-Side-Angle)
Any point on a perpendicular
bisector of a segment is
equidistant from the endpoints
of that segment.
In an isosceles triangle(two
sides congruent), the angles
opposite the congruent sides are
also congruent.
C is exactly equidistant from A
and B.
AAS (Angle-Angle-Side)
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Analytic Geometry Condensed Study Guide
When conducting a proof
involving congruent triangles,
mark the diagram as you go:
4. Perpendicular bisector of a
segment.
8. Square in a circle.
If properly marked, you will see
than reason 6 should be SAS.
There are several constructions
that you will need to know. The
markings are listed below. Go to
5. Construct a perpendicular line
through a point not on the line.
www.mathopenref.com/const
ructions.html for animations.
9. Regular hexagon in a circle.
1. Copy a segment.
6. Construct a parallel line
through a point not on the line.
2. Copy an angle.
3. Bisect an angle.
Unit 2
In this diagram, you originally
had line l and point P.
7. Equilateral triangle in a circle.
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Analytic Geometry Condensed Study Guide
A
These are called trigonometric
functions.
In a right triangle, the two acute
triangles are complementary, or
adds to 90. The sine of an angle
equals the cosine of its
complement an d vice versa.
Unit 3
To find a missing side in a right
triangle with one side and one
angle, use the appropriate trig
function and solve.
A radius of a circle is
perpendicular to a tangent at the
point of tangency.
sin 50 = cos 40
cos 20 = sin 70, etc.
The Pythagorean Theorem
helps find the third side of a
right triangle if you know two
sides. 𝑎2 + 𝑏 2 = 𝑐 2 The
hypotenuse is always c.
To find x, use the ration
𝑥
sin 75 =
12
Two tangents from a common
external point are congruent.
𝑥 = 12 ∙ sin 75
𝑥 = 11.59
𝑠𝑖𝑛𝐴
𝑡𝑎𝑛𝐴 =
𝑐𝑜𝑠𝐴
If you know one trig function,
you can find the other two by
building a right triangle. For
Typically, if the variable is on
top, you multiply. If it’s on the
bottom, you divide.
The angle of depression is 3
degrees in the diagram:
The measure of an arc of a circle
equals its central angle.
3
example, if sin 𝐴 = , draw a
5
right triangle, label one acute
angle A, label the opposite as 3
and the hypotenuse as 5, find the
missing side with the
Pythagorean Theorem, then find
cos or tan.
The angle of elevation is 32
degrees in the diagram:
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Analytic Geometry Condensed Study Guide
The measure of an arc is one
half of an inscribed angle that
intercepts it.
the two intercepted arcs they
face:
When two chords cross in a
circle, the product of each
portion of each chord is equal to
the product from the other
chord:
It follows than an inscribed
angle that intercepts a semicircle
is a right angle.
An angle formed by a tangent
and chord resembles an
inscribed angle and is also half
the intercepted arc:
When any combination of
tangents and secants intersect a
circle, the outside angle is half
the difference of the intercepted
arcs:
When you have tangents or
secants, the portions of segments
obey the rule “outside * whole
= outside * whole.
When two chords intersect
inside a circle, the vertical
angles formed are the average of
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Analytic Geometry Condensed Study Guide
formula 𝐶 = 𝜋𝑑 or 𝐶 =
2𝜋𝑟. To find the arclength of an
𝑚
arc, use the formula 𝐿 =
∙
360
2𝜋𝑟. Notice how you are just
finding a fraction of the
circumference.
The variable B refers to the area
of the base, which is 𝜋𝑟 2 for a
circle.
The opposite angles of an
inscribed quadrilateral are
supplementary:
1
Pyramid: 𝑉 = 𝐵ℎ
3
Angles and arcs can also be
measured in radians.
𝝅 radians= 180 degrees. To
convert from radians to
degrees, multiply by
180
.
𝜋
To
convert from degrees to
𝜋
radians, multiply by 180. An
Prism: 𝑉 = 𝐵ℎ
entire circle is 2𝜋 radians.
Volume measures the amount of
space enclosed by an object.
__________________________
The area of an entire circle can
be found by the formula 𝐴 =
𝜋𝑟 2 . The area of a sector of a
circle can be found with the
𝑚
formula 𝐴𝑆 =
∙ 𝜋𝑟 2 . Notice
Cylinder: 𝑉 = 𝜋𝑟 2 ℎ (𝑉 = 𝐵ℎ)
Pyramids and Prisms can have
any polygonal base.
4
Sphere: 𝑉 = 𝜋𝑟 3
3
360
how you are just finding a
fraction of the area.
1
1
3
3
Cone: 𝑉 = 𝜋𝑟 2 ℎ (𝑉 = 𝐵ℎ)
The entire circumference of a
circle can be found by the
By Cavalieri’s Principle, two
figures have the same volume if
each of their cross sections are
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Analytic Geometry Condensed Study Guide
equal. The cylinders below have
the same volume:
Rational numbers can be
written as the ratio of two
𝑝
integers . The decimal
To multiply polynomials,
distribute:
𝑞
equivalent of any rational
number is a terminating or
repeating decimal. Examples
include:
Unit 4
Some basic root examples:
Rational exponents are
exponents that are fractions.
Below, b is called the index of
the root.
If you add, subtract, multiply, or
divide rational numbers(except
by 0), the result is a rational
number. This is called closure.
Like rational numbers,
polynomials are closed under
addition, multiplication, and
subtraction.
Irrational numbers cannot be
written as the ratio of two
integers. Examples include:
Polynomial operations are
involved with perimeter and
area problems:
If you add, subtract, or multiply
a rational number and an
irrational number, the result is
always irrational.
Perimeter
A polynomial consists of
constants, variables, and
exponents. Examples include:
Some basic exponent rules:
If you add or subtract
polynomials, just combine like
terms. When subtracting, be
careful to change all of the signs
in the second polynomial.
Area
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Analytic Geometry Condensed Study Guide
The imaginary unit is 𝑖 =
√−1. Any number containing 𝑖
is called an imaginary number.
Quadratic equations are
equations of the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑎 ≠ 0.
There are several ways to solve
a quadratic equation:
The powers of i fall in a
repeating pattern:
1. graphing(not efficient unless
graph is given)
𝑖0 = 1
2. square roots (only if b=0)
𝑖1 = 𝑖
3. factoring(only if factorable)
𝑖 2 = −1
4. quadratic formula
𝑖 3 = −𝑖
Methods 2 and 3 are easier than
4, but each is limited as to when
they can be used.
𝑖4 = 1
𝑖5 = 𝑖
Etc.
Complex numbers are written in
the form 𝑎 + 𝑏𝑖. When adding
or subtracting complex numbers,
treat them as if they are
polynomials and as if 𝑖 were a
variable(though it’s not).
Here, 1 and -1 are the two
solutions
Sometimes, the solutions, or
roots, are complex.
Method 1- Graphing
If a graph is given, the xintercepts are the solutions. You
may have 2 real solutions, 1
repeated solution, or 2
complex(0 real) solutions.
There are no x-intercepts. The
solutions are complex and can
not be determined from the
graph.
Method 2- Square Roots
The following illustrates the
method. Notice that b=0 from
𝑎𝑥 2 + 𝑏𝑥 + 𝑐. (No x term)
When multiplying complex
numbers, it is especially helpful
to remember that 𝑖 2 = −1.
(8 + 6𝑖)(2 − 3𝑖)
16 − 24𝑖 + 12𝑖 − 18𝑖 2
16 − 12𝑖 − 18(−1)
Here, the number 0 is a repeated
solution.
Method 3- Factoring
34 − 12𝑖
Unit 5
There are several factoring
techniques.
Always look for a GCF before
you look for a pattern:
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Analytic Geometry Condensed Study Guide
2𝑥 2 + 4𝑥 = 2𝑥(𝑥 + 2)
The vertex can be determined
without completing the square.
Difference of Two Perfect
Squares
2
To find the x-coordinate of the
vertex, use the expression −
2
𝑥 − 𝑦 = (𝑥 − 𝑦)(𝑥 + 𝑦)
𝑥 2 + 6𝑥 + 8 Here 𝑏 = 6 𝑎𝑛𝑑 𝑎 = 1
4𝑎2 − 25𝑏2 = (2𝑎 + 5𝑏)(2𝑎 − 5𝑏)
The x-coordinate is −
𝑥 4 − 36𝑦 4 = (𝑥 2 − 6𝑦 2 )(𝑥 2 + 6𝑦 2 )
7,8, and -9 are called coefficients.
𝑥 2 + 3𝑥 − 18 = (𝑥 + 6)(𝑥 − 3)
7𝑥 2 ,8𝑥, and -9 are called terms.
2𝑥 2 + 15𝑥 + 7
(𝑥 +
14
1
) (𝑥 + )
2
2
(𝑥 + 7)(2𝑥 + 1)
When you have an equation, set
each factor equal to zero and
solve:
7 is called the leading
coefficient.
The degree, or highest power, or
any quadratic equation is 2.
Completing the Square changes a
quadratic equation from standard
form (no parenthesis) to vertex
form (one parenthesis.
𝑦 = 𝑥 2 + 6𝑥 + 8
(𝑥 − 7)(𝑥 − 8) = 0
𝑦 − 8 = 𝑥 2 + 6𝑥
𝑥 = 7 𝑜𝑟 𝑥 = 8
𝑦 − 8 + 9 = 𝑥 2 + 6𝑥 + 9
𝑦 + 1 = (𝑥 + 3)2
𝑦 = (𝑥 + 3)2 − 1
Method 4-Quadratic Formula
To find the zeros of the above
functions, make ℎ(𝑡) = 0 and
solve. You must use the quadratic
formula as you can’t use the other
methods.
=−
−6
2(1)
=
One common application of
quadratic equations are area
problems. For example, a rectangle
has a width 5 more than its length.
Find the dimensions if the area of
the rectangle is 50 𝑖𝑛2 .
x = length, 𝑥 − 5 = width
𝐴 = 𝑙𝑤
-9 is called the constant term.
𝑥 2 − 15𝑥 + 56 = 0
𝑥 − 7 = 0 𝑜𝑟 𝑥 − 8 = 0
𝑏
2𝑎
−3. Plug the -3 back in to get -1.
An example of a quadratic
expression is 7𝑥 2 + 8𝑥 − 9.
𝑥 2 + 9𝑥 + 20 = (𝑥 + 4)(𝑥 + 5)
Trinomials with Leading
Coefficient not 1:
.
Put this value back into the
equation to find y:
𝑥 2 − 9 = (𝑥 − 3)(𝑥 + 3)
Trinomials with Leading
Coefficient 1:
𝑏
2𝑎
The nine comes from taking half
of 6 and squaring it.
The significance of vertex form is
that you can easily determine the
vertex of the parabola. In this case
it’s (-3,-1). Change the sign inside
the parenthesis, but not outside the
parenthesis.
𝑥(𝑥 − 5) = 50
𝑥 2 − 5𝑥 = 50
𝑥 2 − 5𝑥 − 50 = 0
(𝑥 − 10)(𝑥 + 5) = 0
𝑥 = −10 or 𝑥 = 5
Granted that length can’t be
negative, it follows that the length is
5 and the width is 10.
The formula
is also a common application
problem. Typically the variables
𝑣𝑜 (initial velocity) and ℎ𝑜 (initial
height) are provided. -16 is a
constant due to gravity. With metric
units, -4.9 may be used in place of
-16.
How long would it take for a ball
launched with an initial velocity of
48 ft/s from an initial height of 160
feet take to hit the ground? Use the
formula
9
Analytic Geometry Condensed Study Guide
ℎ(𝑡) = −16𝑡 2 + 48𝑡 + 160
ℎ(𝑡) = 0 when the ball is on the
ground.
0 = −16𝑡 2 + 48𝑡 + 160
0 = −16(𝑡 2 − 3𝑡 − 10)
0 = −16(𝑡 − 5)(𝑡 + 2)
You can also graph such a system
and observe their intersection
points:
𝑡 = 5 𝑜𝑟 𝑡 = −2
When the leading coefficient of a
quadratic equation is negative, the
graph is and upside-down u, has a
vertex that is a maximum of the
graph, and first increases then
decreases:
The answer is 5 as time can’t be
negative.
To solve a system of equations
including a linear equation and
quadratic equation, isolate y from
the linear equation and set the result
equal to the quadratic equation and
solve. There are often two
solutions:
The solutions are approximately
(-3,-6) and (2,-1).
To find the average rate of
change for a quadratic equation
over the interval [a,b], apply the
formula 𝑟 =
Get y by itself:
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
. This is
basically the slope formula.
Find the average rate of change
from −1 ≤ 𝑓(𝑥) ≤ 2 using the
table.
Set this expression equal to the
quadratic equation:
𝑥
-1
0
1
2
5
4
5
8
𝑟=
Now plug each x back into the
linear equation to find the y’s that
go with them:
𝑓(𝑥)
A recursive process can show that
a quadratic function has second
differences that are equal to each
other.
Notice the second differences for
the quadratic equation
8−5
3
= =1
2 − (−1) 3
When the leading coefficient of a
quadratic function is positive, the
graph is a u-shape, has a vertex
that is a minimum of the graph,
and first decreases then increases:
A recursive function is one where
each value is based on the
previous one. When working with
multiple choice problems
involving formulas, it is often best
to work backwards:
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Analytic Geometry Condensed Study Guide
Find the equation that matches the
sequence: 8,17,32,53,80,…
a) 3𝑥 2
c) 5𝑥 2
b) 3𝑥 2 + 5
d) 5𝑥 2 + 3
Plug 1 into each formula. 1
should give you 8(first term)
Plug 2 into each formula. 2 should
give you 17(second term).
Multiplying by a number whose
absolute value is larger than one
makes a function vertically stretch
(get more narrow) 𝑓(𝑥) = 3𝑥 2
Etc.
𝑓(𝑥) = 2𝑥 3 − 11𝑥 is an odd
function. Notice that its variable
exponents are all odd(3 and 1).
Notice that 𝑓(2) and 𝑓(−2) are
now opposites. 𝑓(2) = −6
𝑓(−2) = 6.
Remember from coordinate
algebra that exponential functions
may start out small, but their
values will eventually exceed
those of linear or quadratic
functions.
B is the correct answer here….
A graph can be transformed in
several ways.
Multiplying by a number whose
absolute value is less than one
makes a function vertically shrink
Consider 𝑓(𝑥) = 𝑥 2
(get wider) 𝑓(𝑥) = 𝑥 2
1
3
A quadratic regression is a curve
that best fits a set of data. If a set
of data takes on the approximate
shape of a parabola, you can draw
a parabola through the data to
make observations.
Unit 6
Adding outside a function makes
it go up. 𝑓(𝑥) = 𝑥 2 + 1
Multiplying by a number that is
negative will make a graph reflect
over the y-axis (turn upside-down)
𝑓(𝑥) = −𝑥 2
Subtracting inside a function
makes it go down. 𝑓(𝑥)= 𝑥 2 − 1
Even functions have variable
exponents that are all even, are
symmetric to the y-axis, and have
the property 𝑓(𝑥) = 𝑓(−𝑥).
Adding inside a function makes it
go left. 𝑓(𝑥) = (𝑥 + 1)2
Subtracting inside a function
makes it go right. 𝑓(𝑥) =
(𝑥 − 1)2
𝑓(𝑥) = 7𝑥 4 + 9𝑥 2 − 3 is an even
function. Notice that its variable
exponents are all even(4,2,and 0).
If you put any number in, say 2,
its opposite,-2, would yield the
same value if plugged in. 𝑓(2) =
145, 𝑓(−2) = 145
The equation of a circle on a
coordinate plane is
where (h,k) is the center and r is
the radius. For the equation
(𝑥 − 3)2 + (𝑦 + 2)2 = 9, the
center is (3,-2) and the radius is
3.
The area formula for a circle is
derived from the Pythagorean
Theorem.
Sometimes the equation is not
given with parenthesis. In this
case, you have to complete the
square in order to find the
center:
Odd functions have variable
exponents that are all odd, are
symmetric to the origin, and have
the property 𝑓(𝑥) = −𝑓(−𝑥).
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Analytic Geometry Condensed Study Guide
(𝑥 2 ) our answer is
The center is (1,2), the radius
√2.
When the parabola opens
horizontally, the formula is
A parabola is not only the graph
of a quadratic function, but can
be defined as all of the point that
are equidistant from a fixed
point called the focus and a
fixed line called the directrix.
The picture in this situation:
When solving a system of
equations involving a line and a
circle, the method is similar to
that of a line and quadratic.
Find the intersection points of a
line with slope of 2 that passes
through the origin and a circle
with radius √5 that is centered
at the origin.
The system of equaitons:
𝑥 2 + 𝑦2 = 5
𝑦 = 2𝑥
Plug 2𝑥 in for 𝑦:
𝑥 2 + (2𝑥)2 = 5
𝑥 2 + 4𝑥 2 = 5
5𝑥 2 = 5
When the parabola opens
vertically, its formula is
Where (h,k) is the vertex and p
is the distance that the focus and
directrix is from the vertex.
Write the equation of the
parabola whose focus is (6,4)
and whose directrix is 𝑦 = 2.
Since the directrix is a
horizontal line, the graph opens
vertically and the x-coordinate
of the focus and vertex will be
the same. So ℎ = 6. k, the ycoordinate of the vertex, is
halfway between the focus and
directrix. Halfway between 2
and 4 is 3. So 𝑘 = 3. p is the
distance between the directrix(or
focus) and the vertex. The
distance from 3 and 2 is 1, so
𝑝 = 1. Using the correct form
𝑥2 = 1
𝑥 = ±1
Plug both 1 and -1 back into
𝑦 = 2𝑥 to get 2 and -2. So the
solutions are (1,2) and (-1,-2).
Like before, there could be 2
solutions, 1 solution, or no
solutions.
__________________________
To show that a point is on a
circle with given radius and
center, first find the equation of
the circle and then plug the point
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Analytic Geometry Condensed Study Guide
into the equation to see if it
works.
Is (8,2) on the circle whose
center is (5,-2) and whose radius
is 5? Find the equation of the
circle:
(𝑥 − 5)2 + (𝑦 + 2)2 = 25
Plug (8,2) in:
(8 − 5)2 + (2 + 2)2 = 25
32 + 42 = 25
25 = 25
It works, so (8,2) is on the
circle.
__________________________
If asked to show any geometric
property on a coordinate grid,
you may need to use the
distance or midpoint formulas:
{1,2,3,4,5,6}. Rolling a 4 would
be an event as would rolling an
even number.
The intersection”and” of two
events is what the events have in
common. The symbol for the
intersection of events 𝐴 and 𝐵 is
𝐴 ∩ 𝐵. If two events are
independent, then 𝑃(𝐴 ∩ 𝐵) =
𝑃(𝐴) ∙ 𝑃(𝐵).
The union”or” of two events is
all of the outcomes of either
event. They symbol for the
union of events A and B is 𝐴 ∪
𝐵. If there is no overlap, then
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
are juniors, so you get the same
answer:
6
18
1
= .
3
The complement of an event is
the set of events not included in
the favored outcome. In
symbols, the complement of A
is denoted 𝐴′ and is sometimes
read “not A”. For example, the
probability that a randomly
2
selected day starts with “T” is .
7
The complement of this event is
all other days, and the
probability of the complement is
2
5
7
7
1− = .
Use the Venn diagram:
Conditional
Probability, 𝑃(𝐴|𝐵), is the
probability of an event
happening given that another
event has occurred. The formula
for conditional probability is
If A represents those with a
bicycle, and B those with a
skateboard, then
𝑀=
𝐴 ∩ 𝐵 contains Joe, Mike,Linda,
and Rose
Unit 7
Probability=
# 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
The symbol for the probability
of A is 𝑃(𝐴).
A sample space is the set of all
possible outcomes. A subset
from this group is called an
event. For example, the sample
space when rolling a die is
Find the probability that a
randomly selected student would
be a junior granted that the
student owns a car.
Answer:
Notice that in many cases it is
easier just to note that 18
students have a car, 6 of which
𝐴 ∪ 𝐵 contains everyone except
Amy, Gabe, and Abi
(𝐴 ∪ 𝐵)′ contains Amy, Gabe
and Abi.
Two separate events are
independent if the outcome of
one event does not affect the
outcome of the other. If one
outcome does affect the other,
the events are called dependent.
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Analytic Geometry Condensed Study Guide
If two events are independent,
then
𝑃(𝑔𝑖𝑟𝑙 𝑜𝑟 𝑛𝑎𝑚𝑒 ℎ𝑎𝑠 5 𝑙𝑒𝑡𝑡𝑒𝑟𝑠)
4
1
1
4
7
7
7
7
.= + − =
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)
What’s the probability that a
randomly selected day would
start with an S and a randomly
selected month would start with
an A?
Since one event doesn’t affect
the outcome of the other, then
2
the answer is ∙
2
7 12
=
1
21
Another fact about independent
events is that if A and B are
independent, then
𝑃(𝐴|𝐵) = 𝑃(A) and 𝑃(𝐵|𝐴) =
𝑃(𝐵).
Mutually exclusive events have
no overlap. If A and B are
mutually exclusive, then
𝑃(𝐴 𝑜𝑟𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
If two events are not mutually
exclusive, then 𝑃(𝐴 𝑜𝑟 𝐵) =
𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Suppose you have the following
classmates:
Boys: George,John,Thomas,
James
Girls: Martha, Abigail,Dolly
One student is randomly
selected. Find:
P(boy or Martha). There is no
overlap, so P(boy or Martha)=
4
7
1
5
7
7
+ =
𝑃(𝑔𝑖𝑟𝑙 𝑜𝑟 𝑛𝑎𝑚𝑒 ℎ𝑎𝑠 5 𝑙𝑒𝑡𝑡𝑒𝑟𝑠)
. There is overlap here(Dolly is
both) so
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