Standard #: MA.912.DP.4.1

• Within this benchmark, students should use their knowledge of Venn Diagrams and set notation from the Logic and Discrete Theory (LT) benchmarks within their work of these Data Analysis and Probability (DP) benchmarks. The expectation is students are working with multiple events (2 or more). 
• Instruction includes finding the sample space for an event and then subsets of that sample space. 
  • For example, the sample space for rolling a die is S = {1, 2, 3, 4, 5, 6}. The sample space for rolling an even number, which is a subset, is S = {2, 4, 6}. 
• Instruction includes the use of diagrams in order to explore and illustrate concepts of complements, unions, intersections, differences and products of two sets.
  • The intersection of two sets is denoted as A ∩ B and consists of all elements that are both in A and B. 
  • The complement of a set can be denoted as A^c, ^-A^-, A′ or ~A. This is the set of all elements that are in the universal set S but not in A. Complements can be of unions or intersections.
• Instruction includes finding sample spaces for multiple events. 
  • For example, find the sample space of rolling an even number on a six-sided die and flipping a head on a coin. 
    • Sample space for rolling a six-sided die: R = {1, 2, 3, 4, 5, 6}. 
    • Sample space for rolling an even number on a six-sided die: E = {2, 4, 6}. 
    • Sample space for flipping a coin: F = {H, T}. 
    • Sample space for flipping a head: H = {H}. 
    • Sample space for rolling a die and flipping a coin: S = {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}.  
    • Sample space for rolling an even number and flipping a head: A = {2H, 4H, 6H}. 
    • Sample space for not rolling an even number and not flipping a head: B = {1T, 3T, 5T}. 
• Instruction includes finding the sample space of union and intersections. 
  • For example, a deck of 16 cards has 4 red hearts, 4 red diamonds, 4 black spades, and 4 black clubs, numbered 1 to 4. 
    • Sample Space for the deck of cards: S = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S, 1C, 2C, 3C, 4C}.  
    • Sample Space for the intersection of red and 3s: 
      • Define the subset of Red: R = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D}. 
      • Define the subset of 3s: 3s = {3H, 3D, 3S, 3C}. 
      • Define the intersection of Red and 3s: R ∩ 3S = {3H, 3D}. 
    • Sample Space for the union of clubs and not diamonds: 
      • Define the subset of clubs: C = {1C, 2C, 3C, 4C}. 
      • Define the subset of not diamonds: D′ = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S, 1C, 2C, 3C, 4C}.
      • Define the union of clubs and not diamonds: C ∪ D′ = {1H, 2H, 3H, 4H, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S, 1C, 2C, 3C, 4C}.

Common Misconceptions or Errors

• If finding the unions when the events are not mutually exclusive, students often count the intersection more than once because it is included in multiple sets. 
• Students may forget about the values that are not in the sets. 
• Students may confuse the symbols used in basic set notation

Instructional Tasks

• The table below provides information on 10 students within Student Government. For each student, their sex, age, whether or not they plan to go to college, if they play a sport, and how many honors courses they are currently enrolled in.
  
  
  

• Part A. What is the sample space for number of colleges these students plan to apply to? 
• Part B . What outcomes from the sample space in part A are in the event that the student plans to study business? 
• Part C . Consider the following 3 events . For each list , which students would be make up the event: 
  • F = The selected student is female 
  • S = The selected student plays a sport 
  • A = The selected student is less than 18 years old 
• Part D . Based on part C, which outcomes are in the following events? 
  • F ∪ S 
  • (F ∪ S)′
    
  • F ∩ A 
  • Ac

• What is the sample space of not rolling a number greater than 4 on a six-sided die and flipping a head on a coin?