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7th GradeChapter 11
Displaying and Analyzing DataChapter 12
Using Probability
Probability 4/19The result of an actionOutcome
Event An outcome or group of outcomes
Theoretical Probability
Number of favorable outcomes
Number of possible outcomes
Outcome you want
Total outcomes possible
In the name:
Trisha Leanne McDowell
What is the probability of randomly choosing a vowel if the letters were scrambled?
Example
Total outcomes possible (number of letters in name)
Outcome you want (vowels)
7
20Try your name
Finding probabilities from 0 to 1Since probabilities are written as
fractions they can be thought of as between 0 and 1.
A probability of 0 means it would never happen—an impossible event
A probability of 1 means it would always happen—a certain event
0Impossible
½ or 0.5 1Certain
less likely more likely
Suppose you have a spinner with 4 equally spaced colors: red, blue, green, and purple.
What is the probability that the spinner will land on orange?
What is the probability that the spinner will land on blue?
What is more likely, that the spinner will land on blue or green, or that that spinner will land on purple?
Odds
Unfavorable outcomesFavorable outcomesOdds in
Favor
What you wantWhat you don’t want
Example What are the odds in favor of picking a black puppy out of a litter of 12 puppies if 4 puppies are black and your eyes are closed?
Odds
Favorable outcomesUnfavorable outcomesOdds
against
What you don’t wantWhat you want
Example You have a standard 6 sided dice.What are the odds against rolling a 3?What are the odds against rolling a
multiple of 2?
Workbook
Page 199-200
You try
Lists each data item with the number of times it occurred
Frequency Table and Line plots 4/20
Frequency Table
Display the set of data in a frequency table: 1 4 0 3 0 1 3 2 2 4
Example
Number 0 1 2 3 4
Frequency 2 2 2 2 2
Range Difference between the largest and smallest values in a data set
Make a frequency table for the ages of students in this classroom.
1. Determine the range of ages of so you know what ages to list on the table
Age
Frequency
2. Gather data to determine the frequency of each age.
Displays data with an X mark above a number line
Line Plots
Write your favorite number (between 0 and 10) on the scrap of paper given to you
When your number is called come up to the board and place an x above your number—if there is already an x above your number, then put your x above that x
Use the information from the line plot you make a frequency chart on your own paper
Can you think of other data that could be arrange in a frequency chart or line plot?
Workbook
Page 183-184
You try
Mean, Median, and Mode 4/21Average, the sum of the data divided
by the number of data pointsMean
Find the mean: 2, 5, 6, 12, 6, 8, 12Example
2 + 5 + 6 + 12 + 6 + 8 + 127
517
7.29
The mean number of hours middle schoolers watch TV is 5 hours per night. How much TV do they watch in a week?
(Mean # hour)(# of nights)
(5)(7)
35 hours per week
The middle value when the data set is in order from least to greatest
Median
Find the median: 2, 5, 6, 12, 6, 8, 12Example
2, 5, 6, 6, 8, 12, 12 Median: 6
Find the median: 3, 11, 6, 7, 5, 8, 1, 31, 3, 3, 5, 6, 7, 8, 11 5 and 6 share the middle so find the
mean (5 + 6)/2 11/2 5.5
The number in the data set that repeats the most
mode
example Find the mode: 2, 5, 6, 12, 6, 8, 12
6 and 12 share the mode
Workbook
Page19-20You try
Random Samples and Surveys 4/22
A group of objects or peoplepopulation
sample Part of a population
Random Sample
Each member of a population has an equal chance of being selected in the sample
Identify the population and 3 different sample groups
Example
Elections are in November. Pollsters spend a lot of time and money to try and determine who is going to win.
Random sample: calling names out of the phone book
Not random sample: calling registered Republicans or Democrats
A question that does not influence the sample
Biased Questions
Do you prefer sweet, loving doggies or mean, psychotic cats?
Do you prefer cats or dogs
Fair Questions
A question that makes one answer appear better than another
Example
Workbook
Page 191-192`
Estimating Population Size 4/26
Set two proportions equal to each other
Proportional Reasoning
Population size
SamplePopulation
Proportion SamplePopulation
Sample observedPopulation observed
=
Example 1 out of 6 female American High School Students will have a baby before graduation. What does this statistic predict for the current 7th grade class at OHS? Assume there are 35 girls.
16
x35=
1 • 35 = 6x35 = 6x
5.83 = x
Example There are 20 marked sea otters in a costal region. In a survey, marine biologist counted 42 sea otters, of which 12 were marked. How many sea otters are in that area?
1242
20x=
12x = 42 • 2012x = 840
x = 70
Workbook
Page 193-194
You Try
Sample Spaces 4/27
The result of an actionOutcome
Event An outcome or group of outcomes
Sample Space
List of all possible outcomes
Theoretical Probability
Number of favorable outcomesNumber of possible outcomes
Outcome you wantTotal outcomes possible
You cannot always count the possible outcomes
Multiplication can be used
Counting Principle
Multiply the possible outcomes of each event
We use the last four digits of our Social Security Numbers for lots of things. How many unique combinations are possible?
Four digits so four events
1st digit 2nd digit 3rd digit 4th digit• ••10 10 10 10
10000 possible unique combinations
WZZK is running a contest. If you call in and the last four digits of your Social Security Number are randomly generated, you will $10000.
What is the probability of winning?
Outcomes you want (your SS#)
Possible outcomes (all the combinations)1
10000
You try
Workbook
Pages 205-206
Permutations and Combinations 5/3An arrangement where order is importantPermutation
Example Find the number of ways to arrange the three letters in the word CAT in different two-letter groups where CA is different from AC and there are no repeated letters.
#choicesP#eventsNotation
Because order matters, we're finding the number of permutations of size 2 that can be taken from a set of size 3. This is often written 3P2. We can list them as:
CA CT AC AT TC TA
Letter1 Letter23 • 2
6 possibilities
List
Math
We have 10 letters and want to make groupings of 4 letters. Find the number of four-letter permutations that we can make from 10 letters without repeated letters (10P4),
It is unrealistic to make a list
Letter 1 Letter 2 Letter 3 Letter 4
10 • 9 • 8 • 7
5040 possibilities
List
Math
1. 4P2
2. 6P4
3. 9P4
4. 10P8
You Try
An arrangement where order does not matter
Combination
#choicesC#eventsNotation
Combinations are the number of permutations divided by (the number of events factorial)
Formula
#choicesC#events= #choicesP#events
#events!
7! = 7 • 6 • 5 • 4 • 3 • 2 • 1
4!
6 • 5 • 4 • 34 • 3 • 2 •1
15
Factorial n!= n • (n-1) • (n-2) • (n-3) • . . .• 1
7! = 5040
Find 6C4 6P4
Example Find the number of combinations of size 2 without repeated letters that can be made from the three letters in the word CAT, order doesn't matter; AT is the same as TA.
Because order does not matter, we're finding the number of combinations of size 2 that can be taken from a set of size 3. This is often written
3C2. We can list them as:
CA CT AT
2!
# permutations
6
List
Math
2 • 1
62
3
1. 4C2
2. 6C4
3. 9C4
4. 10C8
You Try
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