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Warm Up 1/17 and 1/18

Warm Up 1/17 and 1/18. Expand (a + b) 2 in standard form; hint use Foil or Box method. Expand (a + b) 3 in standard form. How many term does #1 have? #2? How many terms would (a + b) n have?. Entry 1: Counting Throughout Mathematics.

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Warm Up 1/17 and 1/18

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  1. Warm Up 1/17 and 1/18 Expand (a + b)2 in standard form; hint use Foil or Box method. Expand (a + b)3 in standard form. How many term does #1 have? #2? How many terms would (a + b)n have?

  2. Entry 1: Counting Throughout Mathematics • You saw in the last lesson that counting problems arise in many different contexts. • For example, clothing and menu choices, telephone numbers, computer passwords, automobile license plates, club committees, and card games. • The counting methods you learned are used throughout Algebra, Geometry, probability, graph theory and other areas.

  3. Consider the idea “How many”? • Suppose you flip a coin three times. • How many sequences of heads and tails are possible? • How many sequences have exactly two tails? (hint: list them) • What is the probability of getting exactly two tails when flipping a coin three times?

  4. Probability of an Event • Counting methods are often useful when trying to determine probabilities. • This is especially true when there are a finite of outcomes, all of which are equally likely. • In this case, the Probability of an Event A can be defined as: • P(A) = number of favorable outcomes total number of possible outcomes

  5. Thus, when all the outcomes are equally likely, you can determine the probability of an event by counting the number of favorable outcomes and dividing by the number of possible outcomes. • P(A) = number of favorable outcomes total number of possible outcomes

  6. Example 1 • Consider the experiment of rolling 2 dice, one red and one blue, where an outcome is the number of spots showing on each die face up. • Are all the outcomes equally likely? • What is the total number of possible outcomes?

  7. Example 1 • We are still considering the experiment of rolling 2 dice, one red and one blue. • Consider the event of rolling doubles. How many outcomes are favorable to this event? • What is the probability of getting doubles when rolling 2 dice?

  8. Example 2 • In a state lottery, a player fills out a ticket by choosing five “regular” numbers from 1 to 45, with out repetition, and one PowerBall number from 1 to 45. The goal is to match all the numbers with those drawn at random at the end of the week. The regular numbers chosen do not have to be in the same order as those drawn.

  9. Example 2 • How many different ways are there to fill out the ticket?

  10. Example 2 • A player wins the jackpot by matching all 5 regular numbers plus the PowerBall number. (Called Match 5 + 1) • How many way can this ticket be filled out? • What is the probability of being the Match 5 + 1 winner?

  11. Example 2 • A player wins $100,000 by matching all 5 regular numbers but not the PowerBall number. (Called Match 5) • How many way can this ticket be filled out? • What is the probability of being the Match 5 winner?

  12. Multiplication Rule for Independent Events • The Multiplication Rule for independent events states that if A and B are independent, • Then P(A and B) = P(A) X P(B).

  13. Example 3 • Suppose you have the names of 6 boys and 4 girls on slips of paper in a hat. • You draw a slip of paper, note the name and return the slip of paper to the hat. • You then draw again and note the name.

  14. Example 3 • Can you use the Multiplication Rule for independent events to find the probability that the first name is a girl’s name and the second name is a boy’s name? • Explain why or why not.

  15. Example 3 • Compute the probability using the Multiplication Principle of Counting and the definition of probability. • MPC = # of n ways X # of m ways • P(A) = number of favorable outcomes total number of possible outcomes

  16. Example 4 • Suppose again you have the names of 6 boys and 4 girls on slips of paper in a hat. • You draw a slip of paper, note the name but this time you do NOT return the slip of paper to the hat. • You then draw again and note the name.

  17. Example 4 • Can you use the Multiplication Rule for independent events to find the probability that the first name is a girl’s name and the second name is a boy’s name? • Explain why or why not.

  18. Homework • Practice Worksheet • Entry 2 • #’s 1-3

  19. Warm Up 1/19 and 1/20 What is the probability of an event? Under what conditions can you calculate the probability of an event? What is the difference between with replacement and not with replacement in a probabilistic situation?

  20. Homework Entry 2 #1-3

  21. Warm Up 1/24 and 1/25 • Consider the situation of you having a standard deck of playing cards; 52 cards, 4 suits, ♠,♣,♥,♦. • You select 2 cards, one at time, look at it, and replace it. What is the probability of the first card being a ♦ and the second card being a ♥ ?

  22. Warm Up 1/24 and 1/25 • Still considering the situation of you having a standard deck of playing cards; • 52 cards, 4 suits, ♠,♣,♥,♦. • You select 5 cards, one at time, look at it, and hold it. What is the probability of a royal flush of ♦’s ?

  23. Entry 4Combinations and Pascal’s Δ 1/24 and 1/25 Notes and Practice 1/26 and 1/27 Practice and Review 1/30 and 1/31 Quiz

  24. As in #1 of the WU, we used the: Multiplication Rule for Independent Events • The Multiplication Rule for independent events states that if A and B are independent, • Then P(A and B) = P(A) X P(B).

  25. In WU #2 we used theGeneral Multiplication Rule • If A and B are events that are not independent, • Then P(A and B) = P(A) X P(B given A) • Think of drawing 2 names from a hat without replacement. • Find the probability of that the first name drawn is a girl’s name and the second is a boy’s name.

  26. Entry 3 Review of MPC, Permutations, Combinations • As a group you will present one problem from your homework.

  27. Recall……. • (a + b)2 = a2 + 2ab + b2 with coefficients? • (a + b)3 = a3 + 2a2b + 2ab2 + b3 w/ coefficients? • Let’s investigate coefficients to see a connection: • (a + b)0 • (a + b)1 • (a + b)2 • (a + b)3 • (a + b)4 • (a + b)5 • (a + b)6

  28. Pascal’s Triangle • The triangular array of numbers in the last activity is call Pascal’s Triangle. It is named for the French philosopher and mathematician Blaise Pascal. • He explored many of its properties, especially those related to the study of probability.

  29. Rule’s for constructing Pascal’s Triangle In Pascal’s triangle, each number is the sum of the numbers to its upper left and upper right: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 … … … … … … Let’s create our own Pascal’s Triangle! Calculate out to row 10!

  30. Warm Up 1/26 and 1/27 • Compute C(4,2) • Show the calculation with the formula, using the calculator and using Pascal’s Triangle. • Compute C(6,4). • Show the calculation with the formula, using the calculator and using Pascal’s Triangle

  31. Entry 4 cont’d:Combinations and Pascal’s Δ • Describe how to one would find C(7,3) in Pascal’s Δ? • Describe how to one would find C(n,k) in Pascal’s Δ? • Find each of the following: • C(7,3) C(8,7) C(7,4) C(8,1)

  32. Binomial Theorem • Recall: • (a + b)2 = a2 + 2ab + b2 with coefficients 1,2,1. • (a + b)3 = a3 + 3a2b + 3ab2 + b3 what are the coefficients? • 1,3,3,1. • Describe how these coefficients can be found in Pascal’s Δ?

  33. Binomial Theorem • Describe the pattern with the variable a? • (a + b)2 = a2 + 2ab + b2 • Describe the pattern with the variable b? • Describe a and b in (a + b)3 : • a3 + 3a2b + 3ab2 + b3

  34. Binomial Theorem – Example 0 • (x + y)7 • What is n? • What would the coefficients be? • What happens with the variable x? • What happens with the variable y? • Put it all together.

  35. Binomial Theorem • For any positive integer n: • (a + b)n = • C(n,0)an + C(n,1)an-1b + C(n,2)an-2b2 + ….. C(n,k)an-kbk + ……C(n,n-1)abn-1 + C(n,n)bn

  36. Entry 5 Examples 1 – 4 & Homework 5 – 8

  37. Warm Up 1/30 and 1/31 • Expand ( x + y)6 • Expand ( x + 2)6

  38. Classwork/Homework 1 – 8

  39. Pascal’s Triangle • Write out row 6 from Pascal’s Triangle: • 1 6 15 20 15 6 1 • Expand ( x + y)6 , what do you notice about the variables?

  40. Coefficients using Binomial Theorem • Let’s reason to find the term # and coefficient of the a3b2term in (a + b)6. • Using the expanded form (a + b)6. • A. What “term” is a3b2 from Pascal’s triangle? • B. What combination would we compute? C(___,___)?

  41. Combinations and Pascal’s Δ • Let’s use similar reasoning to find the coefficient of the a12b4term in (a + b)16 • What is n for a14b16 in (a + b)n ? • What is the coefficient of a14b16 term in (a + b)30 ?

  42. Combinations and Pascal’s Δ • What is the coefficient of a2b2 term in (a + b)4 • What is the coefficient of a2b4 term in (a + b)6 • Describe where in Pascal’s Δ you can find the coefficient of a2b2 ? • Describe where in Pascal’s Δ you can find the coefficient of any binomial term an-kbk term in (a + b)n .

  43. Find the coefficient of: • a3b2term in (a + 3)5 • a4b2term in (a + 4)6

  44. Pascal’s Triangle • Since we have C(n + 1, k) = C(n, k – 1) + C(n, k) andC(0, 0) = 1, we can use Pascal’s triangle to simplify the computation of C(n, k): k C(0, 0) = 1 C(1, 0) = 1 C(1, 1) = 1 n C(2, 0) = 1 C(2, 1) = 2 C(2, 2) = 1 C(3, 0) = 1 C(3, 1) = 3 C(3, 2) = 3 C(3, 3) = 1 C(4, 0) = 1 C(4, 1) = 4 C(4, 2) = 6 C(4, 3) = 4 C(4, 4) = 1

  45. Agenda 2/1 and 2/2 • Warm Up • Go over homework (homework check ) • Finalize notes and examples on Binomial Theorem • Go through Notebook Entries • Review for Quiz on 2/3 and 2/6 • Notebooks are due on 2/3 and 2/6

  46. Warm Up 2/1 and 2/2 • Expand ( m – 4 ) 5 • Find the coefficient of a6b2term in (2a + 1)8 • Find the 7th term in (x + 2)6

  47. Entry 7 – Homework 1-5

  48. Notebook Entries • Entry 1 – Notes Probability P(A) • Entry 2 – Practice 1-3 • Entry 3 – Practice MPC, Perm, Comb. • Entry 4 – Notes Comb. w/ Pascal’s Δ • Entry 5 – Practice 1-4 CW, 5-8 HW • Entry 6 – Notes Coefficients w/ Binomial Theorem • Entry 7 – Practice 1-5 • Entry 8 – Notes EXPANDing, YAY! • Entry 9 – Probability & Binomial Theorem (REV)

  49. Examples • Expand ( 2m – 4 )5

  50. Examples • Expand ( 2m – 4n )6

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