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Warm up

Warm up. When rolling a die with sides numbered from 1 to 20, if event A is the event that a number divisible by 5 is rolled: i) What is the sample space, S? n(S)? ii) What is the event space, A? n(A)? iii) What is P(A)? i) S = {1, 2, 3, …, 20}, n(S) = 20 ii) A = {5, 10, 15, 20}, n(A) = 4

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Warm up

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  1. Warm up • When rolling a die with sides numbered from 1 to 20, if event A is the event that a number divisible by 5 is rolled: • i) What is the sample space, S? n(S)? • ii) What is the event space, A? n(A)? • iii) What is P(A)? • i) S = {1, 2, 3, …, 20}, n(S) = 20 • ii) A = {5, 10, 15, 20}, n(A) = 4 • iii)P(A) = 4/20 = 1/5 or 0.2

  2. Finding Probability Using Sets Chapter 4.3 – Dealing With Uncertainty Due now: Complete pp. 209-212 #1, 5, 8-10, 12-13 MSIP/Home Learning: p. 228 #1, 2, 4, 7, 8, 10–14, 17

  3. John Venn 1834 -1923 • “Of spare build, he was throughout his life a fine walker and mountain climber, a keen botanist, and an excellent talker and linguist” -- John Archibald Venn (John Venn’s son), writing about his father

  4. A’ A S A Simple Venn Diagram • Venn Diagram: a diagram in which sets are represented by shaded or coloured geometrical shapes.

  5. Set Notation • In mathematics, curly brackets are used to denote a set of items • e.g., A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} • B = {2, 4, 6, 8, 10} • C = {1, 2, 3, 4, 5} • D = {10} • The items in a set are commonly called elements.

  6. A ∩ B S Intersection of Sets • Given two sets, A and B, the set of common elements is called the intersection of A and B, is written as A ∩ B (“A intersect B”). B A

  7. A ∩ B S Intersection of Sets (continued) • Elements that belong to the set A ∩ B are members of set Aand members of set B. • So… A ∩ B = {elements in both A AND B}

  8. Example 1 - Intersection Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} C = {1, 2, 3, 4, 5} B = {2, 4, 6, 8, 10} D = {10} • a) What is A ∩ B? • {2, 4, 6, 8, 10} or B • b) B ∩ C? • {2, 4} • c) C ∩ D? • { } or Ø (the empty set, sounds like the vowel sound in bird or hurt) • http://encyclopedia.thefreedictionary.com/%D8 • d) A ∩B ∩D? • {10} or D

  9. A U B S Union of Sets • The set formed by combining the elements of A with those in B is called the union of A and B, and is written A U B.

  10. A U B S Union of Sets (continued) • Elements that belong to the set A U B are members of set A or members of set B (or both). • So… A U B = {elements in A OR B (or both)}

  11. Example 2 - Union A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B = {2, 4, 6, 8, 10} C = {1, 2, 3, 4, 5} D = {10} • a) What is A U B? • {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} or A • b) B U C? • {1, 2, 3, 4, 5, 6, 8, 10} • c) C U D? • {1, 2, 3, 4, 5, 10} • d) B U C U D? • {1, 2, 3, 4, 5, 6, 8, 10}

  12. Disjoint Sets • If set A and set B have no elements in common (that is, if n(A ∩ B) = 0), then A and B are said to be disjoint sets and their intersection is the empty set, Ø. • Another way of writing this: A ∩ B = Ø

  13. S A B Disjoint Sets (continued) • A Venn diagram for two disjoint sets might look like this:

  14. The Additive Principle • Remember: • n(A) is the number of elements in set A • P(A) is the probability of event A • The Additive Principle for the Union of Two Sets: • n(A U B) = n(A) + n(B) – n(A ∩ B) • P(A U B) = P(A) + P(B) – P(A ∩ B)

  15. The Additive Principle (cont’d) • Alternatively: • n(A ∩ B) = n(A) + n(B) – n(A U B) • P(A ∩ B) = P(A) + P(B) – P(A U B)

  16. Mutually Exclusive Events • A and B are mutually exclusive events if and only if: (A ∩ B) = Ø (i.e., they have no elements in common) • This means that for mutually exclusive events A and B, n(A U B) = n(A) + n(B)

  17. Example 3 • What is the number of cards that are either red cards or face cards? • Let R be the set of red cards, F the set of face cards • If we have “or” we are looking at union • n(R U F) = n(R) + n(F) – n(R ∩ F) • = n(red) + n(face) – n(red face) • = 26 + 12 – 6 • = 32 • What is the probability of picking a red card or a face card from a standard deck? • P(R U F) = 32/52 = 8/13 or 0.62

  18. Example 4 • A survey of 100 students • How many students study English only? French only? Math only? • We need to draw a Venn diagram

  19. E Example 4: what do we know? • n(E ∩ M ∩ F) = 5 5

  20. 5 E Example 4: what else do we know? • n(E ∩ M ∩ F) = 5 • n(M ∩ E) = 30 • Therefore, the number of students in E and M, but not in F is 25. 25

  21. 5 25 E Example 4 (continued) • n(F ∩ E) = 50 • Therefore, the number of students who take English and French, but not in Math is 45. 45 5 • n(E) = 80

  22. 45 5 5 25 E Example 4 – completed Venn Diagram 17 1 2

  23. MSIP / Home Learning • Read through Examples 1-3 on pp. 223-227 (in some ways, Example 1 is very similar to the example we have just seen). • Exercises: p. 228 #1, 2, 4, 7, 8, 10–14, 17

  24. Warm up • What is the number of cards that are either even numbers (2, 4, 6, 8, 10) or clubs? What is the probability of picking such a card from a standard deck? • Use n(E U C) = n(E) + n(C) – n(E ∩ C) • = n(even) + n(clubs) – n(even clubs) • = 20 + 13 – 5 • = 28 • Probability? • P(E U C) = 28/52 = 7/13

  25. Conditional Probability Chapter 4.4 – Dealing with Uncertainty Learning goal: calculate probabilities when one event is affected by the occurrence of another Due now: p. 228 #1, 2, 4, 7, 8, 10–14, 17 MSIP/HL: pp. 235 – 238 #1, 2, 4, 6, 7, 9, 10, 19

  26. Definition of Conditional Probability • The conditional probability of event B, given that event A has occurred, is given by: P(B | A) = P(A ∩ B) P(A) • Therefore, conditional probability deals with determining the probability of an event given that another event has already happened.

  27. Example 1 • Lights 1 and 2 are both green is 60% of the time. Light 1 is green 80% of the time. What is the probability that Light 2 is green given that Light 1 is green?

  28. Example 1 • The probability that it snows Saturday and Sunday is 0.2. The probability that it snows Saturday is 0.8. What is the probability that it snows Sunday given that it snowed Saturday.

  29. Multiplication Law for Conditional Probability • The probability of events A and B occurring, given that A has occurred, is given by P(A ∩ B) = P(B|A) x P(A)

  30. Example 2 • a) What is the probability of drawing 2 face cards in a row from a deck of 52 playing cards if the first card is not replaced? P(A ∩ B) = P(B | A) x P(A) P(1st FC ∩ 2nd FC) = P(2nd FC | 1st FC) x P(1st FC) = 11 x 12 51 52 = 132 2652 = 11 or 0.05 221

  31. Example 3 • 100 Students surveyed • Refer to yesterday’s Venn diagram. What is the probability that a student takes Mathematics given that he or she takes English?

  32. 17 1 45 5 2 5 25 E Example 3 – Venn Diagram

  33. Another Example (continued) • To answer the question, we need to find P(Math | English). • We know... • P(Math | English) = P(Math ∩ English) P(English) • Therefore… • P(Math | English) = 0.3 = 3 or 0.375 0.8 8

  34. MSIP / Homework • Read Examples 1-3, pp. 231 – 234 • p. 235 – 238 #1, 2, 4, 6, 7, 9, 10, 19

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