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Chapter 9: Counting and Binomial Theorem

Entry Task
Assignments and Homework*
Enter the frequency of your data into THIS GOOGLE FORM (

Which sum appeared most frequently from your dice rolling experiment? Does this make sense? Why/why not?
Combine data from the last homework.

List the sample space of the event where we roll 2 dice.

Calculate the proportion of receiving a particular sum of the dice roll.

Basic Counting
JR:  What differences do you expect for the class data for the dice rolling investigation versus your own?
Pg. 551 #2

Compute the proportions for the number of times each total happened for all class' data (periods 1, 3, and 5).  Organize this information in a table and produce an histogram (by hand) of the results.  Compare the class results to your own (i.e. give similarities and differences).
How many ways are there to select six numbers from a group of 49 unique integers?
Counting Basics (Lesson 9.1; pp. 544-546)

JR: Explain how to tell when a situation is a combination rather than a permutation.
Pgs. 552-554 #5, 6
Arthur Dent has twelve shirts and eight pairs of slacks.  How many “outfits” can he make?  Explain in simple language how you arrived at your solution and include a diagram.
Continue Counting Basics (Lesson 9.1; pp. 544-550)

Overview from lesson:
  • Basic Multiplication Principle
  • Permutations
  • Combinations
  • Notation for each.

Notes from Class

JR: An aircraft’s transponder has four digits, each of which can be zero through seven.  Explain in, simple language, how to determine how many "Squawk Codes" a transponder is capable of displaying.

Pgs. 552-554 #10, 11

Review Notes from Class.
The Seahawks have 77 active players on their roster. During regulation play, 11 make up a team of players. How many different teams can be made?
Teaming Up Activity.

JR: Reflect on instances when:
  • order does/doesn't matter.
  • replacement does/doesn't matter.
Pgs. 551-553 #1, 3, 7
At the Boeing plant in Everett, there are about 30,000 people that work in 3 shifts. There are approximately 14 different jobs on the factory floor, but generally people are only eligible to work one of the jobs.

Suppose you were to schedule employees for work, explain how you would plan a schedule for the Aircraft mechanic (total of A eligible employees for a line jobs), the Electricians (E eligible employees for e line jobs) and the Machinists (M eligible employees for m line jobs).
Compound Events (Lesson 9.2; pp. 557-560)

Pg 561#  3, 4, 5, 10, 13

  • Combinations vs Permutations
  • Multiplication principle
  • Addition principle

JR: How many numbers that are greater than 6000 can be formed from the digits 3,4,5,6,7?

p. 561 # 2, 9
A committee is to be formed from a group of eight students: 5 boys and 3 girls.  The committee must have 2 boys and 2 girls.  How many different committees can be formed?
Quiz 9.2 (determine the number of possible outcomes given a "real world" scenario). 3 x 5 note card okay, Calculators not allowed.

JR: Explain the addition and multiplication principles. When would you use each?

Pg. 551 #4; p. 562-565 #6, 8
Consider a string of four 0s and 1s. Use at least two counting techniques to determine the number of different codes that are possible.
Create ten rows of Pascal's Triangle.  See p. 571 #3.

The Binomial Theorem (Lesson 9.3; pp. 566-570).  Expand the binomial (a + b)5

JR:  What is the connection between combinatorics and The Binomial Theorem?
Pg. 571-572 #3, 4, 7.
What is the probability of rolling a sum of six on a standard pair of dice?  What is the probability of NOT rolling a sum of six?
What is the probability of selecting exactly five college graduates (for a jury of twelve) from a population that has 30% college graduates?  What if the number of college graduates were five or less?

Create a probability distribution modeling the selection of jurors from a population that has 30% college graduates.

Record your data in this Google Form

JR:  What is required for a situation to be considered a "binomial probability?"
  1. Graph the probability distribution (Number of College Graduates vs. proportion) for YOUR data, everyone with same proportion and class data for a different proportion. (Check for strange data). These will be histograms! For example, if the population of college graduates I had today was 30%, I would draw a histogram for:
    1. My data from class
    2. All other students who had 30% College Graduates in the population and
    3. Another population that with a different proportion of college graduates from the data table.
  2. Complete the following in order:
  • Expand the binomial (a + b)12
  • Substitute 0.30 for "a" and 0.70 for "b" and use the proper coefficient from Pascal's Triangle.   Simplify--how does this expansion compare to the activity and probability distribution?
(6 period)
Calculate the probability of rolling a total of six or less five times on a pair of dice in twenty rolls.  Could this be considered a "rare event?"

Complete the survey linked HERE
Record your data in this Google Form

Create a histogram of the proportions of the class data from this Google Sheet for the population assigned yesterday.

Learn the Binomial Distribution

Use the Binomial Distribution to create a histogram of the theoretical probability for each outcome.

JR: Write down the binomial distribution, explain the meaning behind each part of the distribution.

1. Finish the theoretical histogram for Jury Selection for your population from Tuesday. Show the proportion for each category.

2. Write the appropriate binomial and expand it to  model the outcome of having three children if the probability of having a boy child is 51%.  Calculate the probability of having
  • Zero boys
  • One boy
  • Two boys
  • Three boys
(6 period)
Explain what happens when the probability of all possible outcomes are added together. Provide an example.
The Cumulative Probability Density Function.  Begin Probabilities Worksheet.

Connections Activity between Binomial Expansion and Probability from a Binomial Distribution.

JR:  Explain how binomcdf can be found from binompdf.
Finish  Probabilities Worksheet.

Begin Chapter Review Problems: Pg. 574 #3, 4, 7, 8, 11 (use Binomial Theorem), 15.
Boston Market boasts it has "over 3000 meals" created by choosing 3 of its 16 side dishes.  Why is this not correct?  Be explicit, then compute the correct number of meals. Problem demonstrations.

JR:  Derive the appropriate term of the binomial (a + b)20 and use it to determine the probability of rolling a total of six seven times on a pair of dice in twenty rolls.  Use this answer to also determine the probability of NOT rolling a total of six on a pair of dice exactly seven times in twenty rolls.
Derive the appropriate term of the binomial that will compute the probability that exactly sixty people will pass a test out of a sample of 105 if the probability of passing is 0.60.

Continue Chapter Review Problems: Pg. 574 #3, 4, 7, 8, 11 (use Binomial Theorem), 15

Calculate the probability of getting a sum of ten or less three or fewer times when a pair of twelve-sided dice are rolled five times.
In-class review: Questions from homework, activities, and quizzes will be answered.  Bring your papers and ask questions!

Class Notes

JR:  Explain two purposes of The Binomial Theorem.  Give examples.
Prepare for tomorrow's test.

Finish Chapter Review Problems. Pg. 574 #3, 4, 7, 8, 11 (use Binomial Theorem), 15
Get ready for the Test! 
  • Move to a seat where you have ample room,
  • Obtain all the materials you need before class starts. No materials will be loaned during class!
  • Seat at most two at the square "cafe tables"
  • Place the paper "blinders" between each pair of people. Put yourself in a positive mental state.

Chapter 9 Test, Part 1.  You may use YOUR calculator and a 3 x 5 note card* (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are also allowed.  Expect questions on
  • Counting, combinations, permutations, and other situations.
  • Basic probability situations.
  • Give a specific term from the expansion of a binomial.
  • Binomial probability situations.
  • Determining if a specified outcome is a "rare event."
*Note cards may not be mechanically reproduced (no photo copies, word processing, etc.).

JR:  Suggest some ways Mr. G's teaching of this unit could be improved.
Search your notebooks from former math classes, or conduct an internet search about circles, triangles, midpoints and distance formulas in Euclidean Geometry. Bring 1 notebook page of your findings.
*Unless otherwise noted, homework is due the next class day.