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Probability

Probability. Unit Essential Question: How does probability relate to the field of statistics and what implications does this have?. Probability. P1. Counting and Probability. How is probability defined?. Activation.

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Probability

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  1. Probability Unit Essential Question: How does probability relate to the field of statistics and what implications does this have?

  2. Probability P1. Counting and Probability How is probability defined?

  3. Activation A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad? How can you be sure you have all the combinations

  4. Probability Defined

  5. Tree Diagrams Steps: Vertically list the choices for the first event. Draw branches for each choice for the second event coming from each choice of the first. Repeat for each event.

  6. Tree Diagrams Example A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad? ranch mushrooms Italian ranch beans Italian Lettuce ranch cheese Italian ranch mushrooms Italian Spinach ranch beans Italian cheese ranch Italian

  7. Fundamental Counting Principle Tree diagrams are helpful, but what if there are many events and many choices? The fundamental counting principle is a mathematical method to figure out the number of ways a compound event may occur. # of outcomes for event 1 *# of outcomes for event 2*…

  8. Fundamental Counting Principle Ex Mr. Giuliano has 2 pairs of shoes, 3 pairs of pants, 10 shirts, and 4 ties. How many different outfits can he make before he repeats?

  9. Permutations Permutations: all the possible ways a group of objects can be arranged or ordered. Order DOES matter! Example: There are four different books to be placed in order on a shelf. A history book (H), a math book (M), a science book (S), and an English book (E). How many ways can they be arranged? H, M, S, E M, E, S, H S, M, E, H E, M, S, H H, M, E, S M, E, H, S S, M, H, E E, M, H, S H, S, E, M M, S, H, E S, H, M, E E, H, M, S H, S, M, E M, S, E, H S, H, E, M E, H, S, M H, E, M, S M, H, E, S S, E, M, H E, S, M, H H, E, S, M M, H, S, E S, E, H, M E, S, H, M

  10. Factorials Sometimes we do not need to use all the objects in a given set EX: Arrange 3 of the 4 books EX: Arrange 2 of the 5 books

  11. Factorials The product of all the numbers from thegiven number down to 1 n! = n (n-1) (n-2) (n-3) ••• 3 (2) (1) Definition 1! = 1 and 0! = 1

  12. Permutation Formula A permutation of n objects r at a time follows the formula Where: n=total objects in the set r=the objects to be used So if we are to arrange 3 of 7 objects….

  13. Homework Worksheet 1

  14. P2 Repetitions and Circular Permutations Essential Question: What is the difference between replacement and repetition?

  15. Activation What does replacement mean—how might that relate to probability?

  16. Replacement Replacement—using the same object again (nr) Example: The keypad on a safe has the digits 1- 6 on it how many: a) four digit codes can be formed _____ _____ _____ _____ b) four digit codes can be formed if no 2 digits can be the same _____ _____ _____ _____

  17. Repetitions Repetition—occurs when you have identicalitems in a group Example: Find all arrangements for the letters in the word TOOL TOOL OLOT LOTO TOLO OLTO LOOT TLOO OTOL LTOO OTLO OOTL OOLT Since the o’s are identical, we eliminate all duplicate possibilities. Hence, there are only 12 arrangements.

  18. Formula for Repetitions where s and t represent the number of times an item is repeated EXAMPLE: How many ways can you arrange the letters in BANANAS

  19. Circular Permutations Circular Permutation—arranging items in a circle when no reference is made to a fixed point ExCircular Permutation—arranging items in a circle when no reference is made to a fixed point Example: How many ways can you arrange the numbers 4 guests around a table? ample: How many ways can you arrange the numbers 1-4 on a spinner? We would expect 4! Or 24 ways but we only have 6 Circular pCircular permutations are always (n-1)! ermutations are always (n-1)! 2 1 1 2 4 4 3 3 1 1 1 1 1 2 2 1 E C A F G? B ? D 3 4 3 4 3 3 2 2 4 2 2 4 4 4 3 3 D

  20. Homework Worksheet #2

  21. P3. Combinations Essential Question: How can you tell the difference between a permutation and a combination?

  22. Activation When a recipe says combine the following ingredients—what does that mean? How is that different than add the following ingredients one at a time?

  23. Combinations Combinations: the number of groups that can beselected from a set of objects The order in which the items in the group are selected does not matter! Notation: The number of combinations of a set of n objects taken r at a time is n=total objects r=number of objects being used

  24. Combinations Example: How many three person committees can be formed from a group of 4 people—Joe, Jim, Jane, and Jill Formula: Joe, Jim , Jill Joe, Jill, Jane Joe, Jim Jane Is Joe, Jane, Jim A different committee? Jim, Jane, Jill

  25. Homework Worksheet Number 3

  26. P4. Probability Essential Question: How is an independent event defined?

  27. Activation How are these two situations different: There are 5 marbles in a bag what is the probability of getting a red one on the second try if three are red and 2 are blue if you choose a marble and replace it? There are 5 marbles in a bag what is the probability of getting a red one on the second try if three are red and 2 are blue if you choose a marble but keep it out?

  28. P4. Probability Independence: When the outcome of one event does not impact the outcome of another event.

  29. Probability If all outcomes are successful, the probability will be 1 If no outcomes are successful, the probability will be 0 So…Probability is 0 ≤ P ≤ 1

  30. Examples What is the probability of getting an ace from a deck of 52 cards? What is the probability of rolling a 3 on a 6 sided die?

  31. More Examples What is the probability of rolling an even number?

  32. More Examples • What is the probability of getting a total of 5 when a red die and green die is rolled?

  33. More Examples • What is the probability of getting 2 spades when 2 cards are dealt at the same time?

  34. Home Work • Worksheet 4

  35. P5. Compound Probability • Essential Question: What is meant by compound probability?

  36. Activation • What is a compound sentence in English? • What about in Algebra? • How might this impact probability?

  37. Compound Probability with “Or” • If events are exclusive, they have no overlap. • Ex. What is the probability of rolling a 3 or a 5? • “Or” implies addition. • P(A or B)=P(A)+P(B)

  38. Compound Probability with “Or” • What is the probability of drawing an ace or a heart? • # of cards_____ • # of hearts____ • # of aces_____ # of hearts that are aces____ • When events are inclusive, they have overlap that must be considered. • OR: P(A or B) = P(A) + P(B) – P(A and B)

  39. Compound Probability with “And” • “And” implies multiplication. • P(A and B)=P(A)*P(B) • Ex. What is the probability of rolling a 4 and drawing an ace? • Are these events dependent or independent?

  40. Homework • Worksheet 5

  41. P6. Binomial Expansion Essential Question: What is binomial expansion and how does it relate to probability?

  42. Activation Look at it in terms a algebra first: Expand (a + b)3 =(a + b)(a + b)(a + b) =(a2 + 2ab + b2)(a + b) =a3 + 2a2b + ab2 + a2b + 2ab2 + b3 =a3 + 3a2b + 3ab2 + b3 What if this had been raised to the 10th power?

  43. Binomial Expansion (a + b)01 (a + b)1 1a + 1b (a + b)2 1a2 + 2ab + 1b2 (a + b)31a3 + 3a2b + 3ab2 + 1b3 The pattern for the variables is simple start with the highest power on the 1st variable and count down, start with 0 on the second variable and count up The pattern on the coefficients is less obvious but it follows Pascal’s triangle • 1 • 1 • 1 2 1 • 3 3 1 • 4 6 4 1

  44. Binomial Expansion n = the exponent r = the position of the term – 1 (for the2nd term r = 1) a = the part in the 1st half of the () b = the part in the 2nd half of the () including the sign Find the 7th term of (4x –y2)9 n = 9 r = 7-1

  45. Examples of Binomial Expansion Find (2x + 3)4

  46. Examples of Binomial Expansion Find (a - b)9

  47. How does this relate to probability? Let a = the probability that an event did not occur b = the probability that it did occur n = the # of trials r = # of success Works for any problem which is a dichotomy—something that either happens or does not happen Since total probability = 1 Then p’ = 1-p

  48. Example of how this relates to probability What is the probability that a family with 9 children has 7 girls? a = .5 b = .5 n = 9 r = 7

  49. Example of how this relates to probability What is the probability that a family with 6 children has at least 4 boys? a = .5 b = .5 n = 6 r = 4 or 5 or 6

  50. Example of how this relates to probability What is the probability that a 300 hitter will hit at least 4 times in 5 hits (a 300 hitter hits 300/1000 times)

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