Sample Space Event of Interest Probability of Obtaining Both Heads 8 Venn

Sample space event of interest probability of

This preview shows page 7 - 15 out of 54 pages.

Sample Space: Event of Interest: Probability of Obtaining Both Heads
Image of page 7

Subscribe to view the full document.

8 Venn Diagram ( 文氏圖 , 維恩圖 ) A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. Event A Ω Probability of an event = AREA of the event in the Venn Diagram 0 Pr 𝐸𝐸 1 Empty Set 𝝓𝝓 : A set without any outcome. Pr 𝜙𝜙 =Area of Empty set = 0 Sample Space 𝛀𝛀 : A set with all outcome Pr Ω =Area of Whole Rectangular area = 1
Image of page 8
9 Union, Intersection and Complement Union ( 聯集 ) 𝐴𝐴 ∪ 𝐵𝐵 : Event that either A or B occurs, or they both occur, (“A unions B”). [A new event] Example: Suppose we roll a fair die once. Event 𝐴𝐴 = an even number = {2,4,6} , Event 𝐵𝐵 = a number greater than 4 = {5,6} , Event 𝐴𝐴 𝐵𝐵 = {2,4,5,6} , Pr 𝐴𝐴 𝐵𝐵 = 4/6
Image of page 9

Subscribe to view the full document.

10 Union, Intersection and Complement Intersection ( 併集 , 並集 , 交集 ) 𝐴𝐴 ∩ 𝐵𝐵 : Event that both A and B occur simultaneously, (”A intersects B”). [A new event] Example: Suppose we roll a fair die once. Event 𝐴𝐴 = an even number = {2,4,6} , Event 𝐵𝐵 = a number greater than 4 = {5,6} , Event 𝐴𝐴 𝐵𝐵 = {6} , Pr 𝐴𝐴 𝐵𝐵 = 1/6
Image of page 10
11 Union, Intersection and Complement [A new event] Complement ( 差集 ) 𝐴𝐴 𝑐𝑐 𝑜𝑜𝑁𝑁 ̅ 𝐴𝐴 : Event that A does not occur (“A complement”). Example: Suppose we roll a fair die once. Event 𝐴𝐴 = an even number = {2,4,6} , Event 𝐴𝐴 𝑐𝑐 = a odd number = {1,3,5} , Pr 𝐴𝐴 𝑐𝑐 = 1 Pr 𝐴𝐴 = 1/2
Image of page 11

Subscribe to view the full document.

12 Properties For any three events, 𝐴𝐴 , 𝐵𝐵 and 𝐶𝐶 , defined on a sample space 𝛀𝛀 Commutativity: 𝐴𝐴 ∪ 𝐵𝐵 = 𝐵𝐵 ∪ 𝐴𝐴 , 𝐴𝐴 ∩ 𝐵𝐵 = 𝐵𝐵 ∩ 𝐴𝐴 Associativity: ( 𝐴𝐴 ∪ 𝐵𝐵 ) ∪ 𝐶𝐶 = 𝐴𝐴 ∪ ( 𝐵𝐵 ∪ 𝐶𝐶 ) ( 𝐴𝐴 ∩ 𝐵𝐵 ) ∩ 𝐶𝐶 = 𝐴𝐴 ∩ ( 𝐵𝐵 ∩ 𝐶𝐶 )
Image of page 12