# EXPERIMENTAL PROBABILITY

```Created by: Miss Jessie Minor
Use: PSSA Review For 7th Grade
Sources: Common Core Standards from PDE website.
Contents: Concepts with description of how to solve and practice
problems.
Reinforcement:
Internet Websites:
www.studyisland.com,
www.ixl.com,
www.mathmaster.org,
and PSSA Coach workbook
EXPERIMENTAL PROBABILITY
IN ORDER TO CALCULATE EXPERIMENTAL
PROBABILITY OF AN EVENT USE THE
FOLLOWING DEFINITION:
P(Event)=
LESSON 30
EXPERIMENTAL PROBABILITY
A student flipped a coin 50 times. The coin landed on heads 28 times.
Find the experimental probability of having the coin land on heads
P(heads) = 28 = .56 = 56%
50
It is experimental because the outcome will change every
time we flip the coin.
Theoretical Probability –The outcome is exact. When we roll a die, the total
possible outcomes are 1,2,3,4,5, and 6. The set of possible outcomes is known
as the sample space.
Find the prime numbers---since 2,3,and 5 are the only numbers in the same
space
P(the number is prime) = 3 = 60%
5
LESSON 29
Rate is comparison of two numbers example: 40 feet per second or 40 ft
1 sec
Unit price = price divided by the units
Sales Tax= change sales tax from a percent to a decimal and the multiply it times the
amount. Finally add that amount to the total to find the total price.
Example 1: \$1200 at 6% sales tax = 100 6
= .06 x 1200 = 72 = \$1272
Example 2: Rachel bought 3 DVDs. Using the 6% sales tax rate, calculate the amount
of tax she paid if each DVD costs \$7.99?
LESSON 4
Distance formula = distance = rate x time
OR
D = rt
Example 1: A car travels at 40 miles per hour for 4 hours. How far did it travel?
d=rt
d=40 miles /hr x 4 hrs
d = 160 miles.
We can also use this formula to find time and rate. We just have to manipulate the
equation.
Example 2: A car travels 160 miles for 4 hours. How fast was it going?
d = rt
160 miles = r (4 hours)
160 miles
4 hrs = t
40 miles/hr = t
LESSON 23
Michael enters a 120-mile bicycle race. He bikes 24 miles an hour. What is
Michael's finishing time, in hours, for the race?
A
B
C
D
2
5
0.2
0.5
DISTANCE =
RATE X TIME
WITH THIS FORMULA WE CAN FIND THE DISTANCE IF THEY GIVE US THE RATE AND
THE TIME. IN FACT, AS LONG AS THEY GIVE ME ANY TWO QUANTITIES, WE CAN FIND
THE THIRD.
EXAMPLE: HOW FAR DID ED TRAVEL IN 7 HOURS IF HE WAS GOING 6O MILES
PER/HOUR
D = RT
D = 60MILES/HR X 7 HRS
D = 420 MILES
OR IF THE DISTANCE IS GIVEN AND THE RATE OR SPEED IS ALSO GIVEN,
D = RT
420MILES = 60 MILES/HR X T
420 MILES
60MILES/HR
=
7 HOURS
Ratio = comparison of two numbers.
Example: Johnny scored 8 baskets in 4 games.
The ratio is 8 = 2
4 1
2 ratios separated by an equal sign . If Johnny score 8 baskets in 4 games how many
baskets will he score in 12 games?
Set up the proportion---
4 games
12 games
Cross multiply
4x = 8 ( 12 )
4x = 96
X= 96
4
LESSON 7
FRACTIONS:
ADDING AND SUBTRACTION ---FIND COMMON DENOMINATORS. Use factor trees, find prime
factors , circle ones that are the same circle the ones by themselves. Multiply the circled
numbers.
Ex ample: 5 + 8
12
9
12
2
2
9
6
3 3
2 3
2 2 2 3
3
3
3 x 3 x 2 x 2 = 36
Common denominator = 36
3 x5 = 4 x 8 = 15 + 32 = 47
36
36
36 36
36
LESSON 1
Multiplying fractions : cross cancel and multiply straight across
4 X 5
5
8
=
1
2
Dividing fractions : change the sign to multiply and reciprocate
the 2nd fraction
3 &divide; 5
4
8 =
3 X 8
4
5
= 24
20
LESSON 2
3 X 5
4
6
1 X 7
49
5 X
9
13
4
5
LCM = least common multiple = the smallest number that 2 or more numbers
will divide into.
Example: find the lcm of 24 and 32
You can multiply each number by 1,2,3,4… until you find a common multiple
which is 96.
Or you can use a factor tree:
24
2 12
2 2 6
2
2 2 3
24:
32:
2 2 2 3
2 2 2 2 2
32
2 16
2 2 8
2 2 2 4
2 2 2 2 2
2x2x2x3x2x2 = 96
GCF = GREATEST COMMON FACTOR = The Largest factor that will divide two or
more numbers. In this case we would multiply the factors that are the same.
2x2x2 = 8, so 8 is the GCF of 24 and 32.
What is the greatest common factor (GCF) of 108 and 420 ?
A
B
C
D
6
9
12
18
What is the least common multiple (LCM) of 8, 12, and 18 ?
A
B
C
D
24
36
48
72
DISTRIBUTIVE PROPERTY
A(B + C) =
AB + AC
We distributed A TO B AND C
Solving 2 step equations:
4(x + 2) = 24
4x + 8 = 24
sub 8
4x = 16
divide by 4 x = 4
Remember when solving 2 step equations do addition and subtraction first then do
multiplication and division first. Just the opposite of (please excuse my dear aunt
sally,) which we us on math expressions that don’t have variables
LESSON 20
Associative and Commutative property
Associative
Commutative
• Always has parentheses
• A ( B X C) = B (C X A)
• FOR MULTIPLICATION
• A + (B + C) = B + (C + A)
• http://www.mathmaster.org/v
ideo/associative-property-formultiplication/?id=932
• AXB=BXA
• FOR MULTIPLICATION
• A+B=B+A
http://www.mathmaster.org/video/com
Stem and leaf plotsBox and –Whisker plots
Investigation 4
LESSON 24
To organize scores or large groups of numbers, we can use stem and leaf plots.
Example
40, 30, 43, 48, 26, 50, 55, 40, 34, 42, 47, 47, 52, 25, 32, 38, 41, 36, 32,
21, 35, 43, 51, 58, 26, 30, 41, 45, 23, 36, 41, 51, 53, 39, 28
Stem
2
3
4
5
Leaf
1 3 5 6 68
0022456689
001112335778
0112358
Stem
2
3
4
5
Leaf
1 3 5 6 68
0022456689
001112335778
Upper quartile
0112358
Lower quartile
MODE—The number that occurs the most often—The mode of these 35 scores is 41.
RANGE—The difference between the least and greatest number—is 37
MEDIAN—is the set of numbers is the middle number of the set when the numbers
are arranged in order—it is 40
MEAN– another name for average is mean
FIRST QUARTILE OR LOWER QUARTILE—The middle number of the lower half of
scores. 32
THIRD QUARTILE OR UPPER QUARTILE—The middle number of the upper half of
scores. 47
LESSON 27, 25
Make a stem and leaf plot from the following numbers. Then make a box and
whiskers diagram.
25, 27, 27, 40, 45, 27, 29, 30, 26, 23, 31, 35, 39
Below are the number of points John has scored while playing the last
14 basketball games. Finish arranging John’s points in the stem and
leaf plot and then find the range, mode, and median.
Points: 5, 14, 21, 16, 19, 14, 9, 16, 14, 22, 22, 31, 30, 31
Stem
Leaf
Range:
0
Mode:
1
Median:
2
3
Lower
extreme
20
Box-and –whisker plot
First quartile Second
Third quartile Upper
quartile
or
or lower
or upper
extreme
median
quartile
quartile
30
40
Inter
quartile
range
50
60
ABSOLUTE VALUE = the number itself without the sign.
The symbol for this is-----
The absolute value of -5
The absolute value of
5
Is 5
Is 5
ORDER OF OPERATIONS
Please, Excuse, My, Dear, Aunt, Sally
3(4 + 4)
&divide; 3-2
3(8)
&divide; 3-2
24
&divide; 3-2
8
-
2
=6
Note that there are not any variables is the statement.
This is why we use order of operation instead of the Distributive property.
LESSON 5
3 + 2(4 x 3)
12 - 15 - 3
(22 + 14) – 6
64 – 8 + 8
http://www.mathmaster.org/video/exponent-properties-involving-products/?id=1889
2&sup3;
3⁴
=
2x2x2
144
= 3x3x3x3
64
4&sup2;
=
4x4
Finding the missing side of a triangle.
Since the sum of the degrees of a
triangle is 180 degrees we subtract the
sum of
65 + 50 = 115 from 180 - 115 = 65
So b = 65
a
50&deg;
65&deg;
b c
If b = 65 to find c we know that a
straight line is 180 so if we subtract
65 from 180 we get 115. Angle c = 115
To find L a we do the same thing.
180 – 50 = 130 so a = 130 degrees.
Pythagorean Theorem
To find the missing hypotenuse of a right triangle, we use the formula
c&sup2;
=
A&sup2;
+
B&sup2;
Hypotenuse
Height = 6 in
c&sup2;
= A&sup2; + B&sup2;
C&sup2; = 6&sup2;in + 8&sup2;in
C&sup2; = 36 sq in + 64 sq in
C&sup2;
= 100 sq in
=
C
sq in
= 10 sq in
Base = 8 inches
http://www.mathmaster.org/video/pythagorean-theorem/?id=1922
AREA OF A TRIANGLE
A + base x height
2
Area = base x height
2
A = 10in x 8 in
2
Height= 8 in
A = 80 sq in
2
A =
40 sq in
Base= 10 in
Definition of height is a line from the opposite vertex perpendicular to the base.
LESSON 12
FINDING AREA OF A TRIANGLE
AREA = &frac12; (BASE X HEIGHT)
A = &frac12; bh
2 ft
height
Area = &frac12; bh
A = &frac12; (4ft)(2ft)
A = &frac12; 8ft
A =4 ft&sup2;
base
4 ft
Finding area of a parallelogram
h
b
Area = b x h
Area of a rectangle = length x width
Area of a square
= side x side
2ft
4ft
2ft
2ft
FINDING PERIMETER AND AREA OF COMPOUND FIGURES
PERIMETER IS THE DISTANCE AROUND A FIGURE.
9 FT
3FT
P = a + b + c + …..
P = 9FT + 9FT + 3FT + 3FT
P = 24 FT
TO FIND THE AREA OF A COMPOUND
FIGURE, ALL WE HAVE TO DO IS FIND THE
AREA OF BOTH FIGURES AND ADD THEM.
6FT
3FT
7FT
2FT
AREA = LENGTH X WIDTH
A = 3FT X 6FT
A = 18FT&sup2;
AREA = LENGTH X WIDTH
A = 4FT X 4FT
A = 16 FT&sup2;
Volume = Length x Width x Height
3 ft
4 ft
5 ft
Parallel lines = lines that never touch--- symbol
Perpendicular lines = lines that intersect---symbol
Skew lines = lines in different planes that never intersect
Plane = many points that are next to each other extending in the same direction
Vertical angles = angles that share a point and are equal---
Adjacent angles = are angles that are 180 degrees and share a side.
Lesson 18
THAT SHARE A COMMON SIDE.
ANGLES 3 AND 4 ARE
ANGLES 2 AND 3 ARE
ANGLES.
2
3
1
4
Complementary angles : angles whose sum =‘s 90 degrees
Supplementary angles: angles whose sum =‘s 180 degrees
Right angle: angle measures 90 degrees ---symbol--Acute angle: angle less than 90
Obtuse angle: angle greater than 90 degrees
Congruent: when two figures are exactly the same
Similar: when two figures are the same shape but not the same size
Regular: when a figure has all = sides
Line of symmetry: when a line can cut a figure in two symmetrical sides
LESSON 17
SUPPLEMENTARY ANGLES
 SUPPLEMENTARY ANGLES ARE ANGLES WHOSE
SUM IS 180 DEGREES.
 COMPLEMENTARY ANGLES ARE ANGLES
WHOSE SUM IS 90 DEGREES.
 A STRAIGHT ANGLE IS EQUAL TO 180 DEGREES
CLASSIFY LINES
INTERSECTING LINES---OCCUPY THE SAME PLANE. THEY
MEET AT ONLY ONE POINT.
PERPENDICULAR LINES WHEN TWO LINES INTERSECT AND
FORM4 RIGHT ANGLES. THE SYMBOL IS ∏
PARALLEL LINES EXTEND FOREVER IN BOTH DIRECTIONS IN
THE SAME PLANE AND NEVER INTERSECT. THES SYMBOL IS
//
SKEW LINES ARE A PAIR OF LINES THAT ARENOT PARALLEL
BUT NEVER INTERSECT. THEY OCCUPY TWO DIFFERENT
PLANES.
Congruent angles and sides mean that they have
the same measure.
Similar figures
Two figures are similar if they have exactly
the same shape, but may or may not have the
same size. The symbol ≈
Points on a coordinate grid
Point of Origin [0, 0]
y
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
[2, 6]
Ordered pair:
2 is x value and
6 is y value
x
axis
LESSON 16
What is the total number of lines of symmetry that can be drawn on the trapezoid below?
A
B
C
D
4
3
2
1
Which figure below correctly shows all the possible lines of symmetry for a square?
A
Figure 1
B
Figure 2
C
Figure 3
D
Figure 4
Chord = line that cuts the circle and
doesn’t go through the center of
the circle.
Diameter = distance across the center of the
circle
Radius = the distance half way across the
circle.
Central angles = angles that are in
the center of the circle
Inscribed angle = the angle on the inside of the
circle
Area = ∏ x r
Circumference = distance around the outside of
the circle
2
Circumference= 2∏r
LESSON 15
CIRCUMFERENCE IS THE DISTANCE
AROUND
--------------------------------
-----------------
DIAMETER- CUTS THE CIRCLE IN HALF IN
THE MIDDLE OF THE CIRCLE
CHORD CUTS THE CIRCLE
ANYWHERE ELSE OTHER THAN
THE MIDDLE
OF THE CIRCLE TO THE OUTER
MOST EDGE
CIRCUMFERENCE AND AREA OF A CIRCLE
CIRCUMFERENCE IS THE DISTANCE
AROUND
--------------------------------
-----------------
OF THE CIRCLE TO THE OUTER
MOST EDGE
DIAMETER- CUTS THE CIRCLE IN HALF IN
THE MIDDLE OF THE CIRCLE
CHORD CUTS THE CIRCLE
ANYWHERE ELSE OTHER THAN
THE MIDDLE
C = ∏ X D
A = ∏ X R&sup2;
LESSON 13
A duck swims from the edge of a circular pond to a fountain in the center of the
pond. Its path is represented by the dotted line in the diagram below.
What term describes the duck's path?
A
chord
C diameter
D central angle
Functions: inserting a value in for x to find y
Example: f(x) = 2x + 4
If x = 2
Then f(x) = 2 (2) + 4
f( x) = 4 + 4
f(x) = 8
So y = 8
Another explanation is-- a function is when we put a value in
LESSON 20
Scientific notation -- 4.057 x 10⁶
Means we are going to move
the decimal 6 places to the
right
4.057 x 10⁶ becomes 4057000
Expanded notation --- numbers written using powers of 10
Example -----4234 = (4 x 10&sup3;) + (2 x 10&sup2;) + (3 x 10&sup1;) + (4 x 10⁰)
4000 + 200 + 30 + 4 = 4234
Any number raised to the zero power = 10 ⁰ = 1
Any number raised to the 1st power = that number
METRIC SYSTEM
KILO
DEKA
METER
LITER
GRAM
HECTO
MILLI
DECI
CENTI
START where your at and move the decimal to where you want to go.
Example:
4 kilometers = 400 meters
LESSON 11
METRIC CONVERSION
KING
HENERY
DIED
DRINKING
KILO
HECTO
DEKA (BASIC ) DECI
UNIT
EXAMPLE
24 KILOGRAMS = _______GRAMS
CHOCOLATE
CENTI
MILK
MILLI
IN 24, THE DECIMAL IS AFTER THE 4 –WE WILL MOVE IT 3 PLACES TO THE
RIGHT AND GET 24,000 GRAMS
Weight Unit Conversions:
USE THE CHART AND MOVE THE DECIMAL POINT.
GRAM = WEIGHT
METER = DISTANCE
LITER = VOLUME
FOR U.S. CUSTOMARY MEASUREMENT, CONVERSIONS ARE ON CHARTS.
The flower box in front of the main city library weighs 124 ounces. What does the
flower box weigh in pounds?
Unit multipliers
Always list the conversion.
For example: change 240 feet to yards
First we list the conversions
There are 3 ft
1 yard
or
1 yard
3 feet
Because I want to go to yards I am going to multiply by 1 yard
3 feet
So 240 feet X 1 yard
240 feet X 1 yard
1
3 feet =
1
3 feet = 80 yards
LESSON 9
Irrational Numbers:
Which of these is an irrational number?
Inequalities: is a mathematical sentence
with one of these symbols &lt;, &gt;, &lt;, &gt;
&lt;
&lt;
&gt;
&gt;
Is less than
Is less than or equal to
Is greater than
Is greater than or equal to
Scaling: A SCALE IS THE RATIO OF THE MEASUREMENTS OF A
DRAWING,A MODEL,A MAP OR A FLOOR PLANTO THE ACTUAL
SIZE OF THE OBJECTS OR DISTANCES
EXAMPLE: AN ARCHITECT’S FLOOR PLAN FOR A MUSEUM EXHIBIY
HALF USES A SCALE OF 0.5 INCH : 2 FEET. On this drawing, a
passageway between exhibits is represented by a rectangle 3.75
inches long. What is the actual length of the passageway?
To find an actual length from a scale drawing, identify and solve a proportion.
Drawing = Drawing
Actual
Actual
Let p = the actual length in feet of the passageway .5 inch
2 feet
.5 = 3.75
.5 x p = 2 x 3.75
2
p
p = 15
http://www.mathmaster.org/video/scale-and-indirectmeasurement/?id=1858
Use cross products to
solve the proportion
LESSON 14
RATIO IS A COMPARISON BETWEEN
TWO NUMBERS.
TWO RATIOS SEPERATED BY AN = SIGN
IS CALLED A PROPORTION.
⅘
= 2/X
TO SOLVE A PROPORTION
WE CROSS MULTIPLY AND
WE GET 4X = 10
X = 10/4
LESSON 7
Use proportions to enlarge or reduce
Mindy and her sisters want to make an enlargement of photograph of their
parents. The original photograph is 5 inches long and 3 inches wide. They would
like the enlargement to be 35 inches long. What is the scale factor of the
enlargement?
The original length is 5 inches and the enlargement length is 35 inches. As a ratio:
Enlargement length /35 inches, 7/original length
or 5 inches / 1
Use a proportion to find the width of the enlargement.
Let x represent the width of the enlargement.
1 ∙ x=7 ∙ 3
Scale factor
X = 21
7/1 = x/3
LESSON 8
Scaling
Scaling is the ratio of the measurements of a drawing, a model, a map, or a
floor plan to the actual size of the objects or distances. A scale drawing is
similar in shape to the object it represents.
Problem—an architect’s floor plan for a museum exhibit hall uses a scale of .5 inch : 2
feet. On this drawing, a passageway between exhibits is represented by a rectangle
3.75 inches long. What is the actual length of the passageway?
To find the actual length from a scale drawing, identify and solve a proportion,
Identify two ratios in the same order
Drawing
= Drawing
Actual
Actual
.5/2 = 3.75/p
.5p = 3.75(2)
.5P = 7.2
P = 15 feet
Rules
RULES FOR MULTIPLYING POSITIVE AND NEGATIVE NUMBERS
NEGATIVE TIMES A NEGATIVE = POSITIVE
POSITIVE TIMES A POSITIVE = POSITIVE
POSITIVE TIMES A NEGATIVE = NEGATIVE
-2 X -4 = 8
2 X 2 = 4
2 X -3 = -6
WHEN WE HAVE A SITUATION LIKE
-(- 2 ) OUR ANSWER IS POSITIVE
2 BECAUSE A NEGATIVE TIMES A
NEGATIVE IS A POSITIVE.
MULTIPLYING AND DIVIDING MIXED NUMBERS
WHENEVER WE MULTIPLY OR DIVIDE MIXED
NUMBERS, ALWAYS CHANGE THEM TO
IMPROPER FRACTIONS
1 3/4 X 11/2 =
7
X 3 = 21
4
2
8
WE DO NOT HAVE TO CHANGE THEM WHEN
Histogram = is a bar graph without the spaces between the bars
6
5
4
3
2
1
4-5 6-7
8-9
10-11
Bar graph looks like this
Spaces between the bars to show difference in data.
LESSON 26
SOLVE PROBLEMS USING PATTERNS
EXAMPLE:
ERIN IS COLLECTING PLASTIC BOTTLES. ON MONDAY SHE HAS 7
BOTTLES, ON TUESDAY SHE HAS 14 BOTTLES, ON WEDNESDAY SHE HAS
21 BOTTLES, AND ON THURSDAY SHE HAS 28 BOTTLES. IF THE PATTERN
CONTINUES, HOW MANY BOTTLES WILL SHE HAVE ON FRIDAY?
1 FIND THE PATTERN TO SOLVE THE PATTERN
2 7,14,21,28
3 WRITE THE DIFFERERENT OPERATIONS THAT YOU CAN PERFORM
ON 7 TO GET 14
4 CHECK THESE OPERATIONS WITH THE NEXT TERM IN THE PATTERN
5 14 + 7 =21
6 14 X 2 = 28
7 FIND THE NEXT TERM IN THE PATTERN TO DETERMINE HOW MANY
BOTTLES ERIN WILL HAVE ON FRIDAY
8 28 + 7 = 35
LESSON 19
SOLVING ONE STEP EQUATIONS
TO SOLVE AN EQUATION, YOU NEED TO GET THE
VARIABLE ALONE ON ONE SIDE OF THE EQUALS
SIGN, YOU CAN USE A MODEL OR AN INVERSE
OPERATION TO SOLVE A ONE STEP EQUATION.
DIVIDE BY 3
3X = 24
3X = 24
3
3
DO IT TO BOTH SIDES
X = 8
LESSON 21
DIVIDING FRACTIONS
RULE---- CHANGE THE SIGN TO MULTIPLY AND RECIPROCATE THE
SECOND FRACTION.
3
4
&divide;
3
4
X
1
1
X
2
2
6
8
=
8
6
2
2
=
1
A square has 4 angles which each
measure 90 degrees
A
D
45
45
45
45
C
B
Find the measure of &lt;A in the
triangle ABC
A
30
B
C
M&lt;A + 90 + 30 = 180
M&lt;A = 60
Modeling Mathematical Situations
Translate “five more than” means 5 plus a quantity
Translate “three times a number” means 3 x n, or 3n
When you combine both you get
5 + 3n or 3n +5
Lesson 22
Comparing and Ordering Integers
NEGATIVE
-6 -5
-4 -3 -2
-1
POSITIVE
0
1
2
3
4
5
-4 IS GREATER THAN -6
LESSON 3
Rational Numbers On a Number Line
Fraction
3
4
Decimal
Percent
.75
4 3
.75 x 100 =
100%
Rational numbers are numbers that can be expressed as fractions that
can be formed from Integers.
Lesson 4
Estimation = Find compatible numbers and divide.
There are 52 weeks in a year. Leo’s salary is \$51,950. \$51,950 is
Divide the compatible numbers.
\$52,000 divided 52 = \$1,000
Lesson 10
Double and Triple Bar Graphs and Double and Triple Line Graphs are used to show
two sets of related data
6
5
4
Series 1
3
Series 2
Series 3
2
1
0
Category 1 Category 2 Category 3 Category 4
Lesson 25
Making Predictions– You can use trends or patterns you see in
graphs to make predictions.
6
5
4
Series 1
3
Series 2
Series 3
2
1
0
Category 1
Category 2
Category 3
Category 4
Lesson 31
```