Chapter 4 Review MDM 4U

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Chapter 4 ReviewMDM 4U

• Gary Greer

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4.1 Intro to Simulations and Theoretical
Probability

• be able to design a simulation to investigate the
  experimental probability of some event
• ex design a simulation to determine the
  experimental probability of more than one of 5
  keyboards chosen in a class will be defective if
  we know that 25 are defective
• get a shuffled deck of cards, choosing clubs to
  represent the defective keyboards
• choose 5 cards and see how many are clubs
• repeat a number of times and calculate probability

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4.2 Theoretical Probability

• work effectively with Venn diagrams
• ex create a Venn diagram illustrating the sets
  of face cards and red cards

S 52
red face 6
red 20
face 6
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4.2 Theoretical Probability

• calculate the probability of an event or its
  complement
• ex what is the probability of randomly choosing
  a male from a class of 30 students if 10 are
  female?
• P(A) n(A)/n(S) 20/30 0.666
• 66.6

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4.2 Theoretical Probability

• ex calculate the probability of not throwing a
  four with 3 dice
• there are 63 possible outcomes with three dice
• only 3 outcomes produce a 4
• probability of a 4 is
• 3/63
• probability of not 4 is
• 1- 3/63

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4.3 Finding Probability Using Sets

• recognize the different types of sets
• utilize the additive principle for unions of sets
• The Additive Principle for the Union of Two Sets
• n(A U B) n(A) n(B) n(A n B)
• P(A U B) P(A) P(B) P(A n B)
• calculate probabilities using the additive
  principle

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4.3 Finding Probability Using Sets

• ex what is the probability of drawing a red card
  or a face card
• ans P(A U B) P(A) P(B) P(A n B)
• P(red or face) P(red) P(face) P(red and
  face)
• 26/52 12/52 6/52 32/52 0.615

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4.4 Conditional Probability

• calculate a probability of events A and B
  occurring, given that A has occurred
• use the multiplicative law for conditional
  probability
• ex what is the probability of drawing a jack and
  a queen in sequence, given no replacement?
• 4/52 x 4/51

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4.4 Conditional Probability

• 100 Students surveyed

• a) Draw a Venn Diagram that represents this
  situation.

• b) What is the probability that a student takes
  Mathematics given that he or she also takes
  English?

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4.4 Conditional Probability
17
45
1
5
2
5
25
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4.4 Conditional Probability

• To answer the question in (b), we need to find
  P(MathEnglish).
• We know...
• P(MathEnglish) P(Math n English)
• P(English)
• Therefore
• P(MathEnglish) 30 / 100 30 x 100 3
• 80 / 100 100 80 8

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4.5 Tree Diagrams and Outcome Tables

• a sock drawer has a red, a green and a blue sock
• you pull out one sock, replace it and pull
  another out
• draw a tree diagram representing the possible
  outcomes
• what is the probability of drawing 2 red socks?
• these are independent events

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4.5 Tree Diagrams and Outcome Tables

• Mr. Greer is going fishing
• he finds that he catches fish 70 of the time
  when the wind is out of the east
• he also finds that he catches fish 50 of the
  time when the wind is out of the west
• if there is a 60 chance of a west wind today,
  what are his chances of having fish for dinner?
• we will start by creating a tree diagram

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4.5 Tree Diagrams and Outcome Tables
P0.3
fish dinner
0.5
west
0.6
0.5
P0.3
bean dinner
fish dinner
0.7
P0.28
0.4
east
bean dinner
0.3
P0.12
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4.5 Tree Diagrams and Outcome Tables

• P(east, catch) P(east) x P(catch east)
• 0.4 x 0.7 0.28
• P(west, catch) P(west) x P(catch west)
• 0.6 x 0.5 0.30
• Probability of a fish dinner 0.28 0.3 0.58
• So Mr. Greer has a 58 chance of catching a fish
  for dinner

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4.6 Permutations

• find the number of outcomes given a situation
  where order matters
• calculate the probability of an outcome or
  outcomes in situations where order matters
• recognizing how to restrict the calculations when
  some elements are the same

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4.6 Permutations

• ex in a class of 10 people, a teacher must
  choose 3 for an experiment (students are done in
  a particular order)
• how many ways are there to do this?
• ans P(10,3) 10!/(10 3)! 720?
• ex how many ways can 5 students be arranged in a
  line?
• ans 5!
• ex how many ways are there above if Jake must be
  first?
• ans (5-1)! 4!

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4.6 Permutations

• ex what is the chance of opening one of the
  school combination locks by chance?
• ans 60 x 60 x 60

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4.7 Combinations

• find the number of outcomes given a situation
  where order does not matter
• calculate the probability of an outcome or
  outcomes in situations where order does not
  matter
• ex how many ways are there to choose a 3 person
  committee from a class of 20?
• ans C(20,3) 20!/((20-3)!3!)

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4.7 Combinations

• ex from a group of 5 men and 4 women, how many
  committees of 5 can be formed with
• a. exactly 3 women
• b. at least 3 women
• ans a
• ans b