Student Actions

Teacher Actions

• Develop mathematical reasoning by exploring and developing reasoning strategies for how to distinguish between and/or represent properties of unions (OR), intersections (AND), and complements (NOT) for a statistical event.

• Develop accurate and appropriate procedural fluency when applying efficient procedures to appropriately model intersections, unions, and/or complements of various statistical events with Venn Diagrams and accurately calculate probabilities of these events. 

• Pose purposeful questions for students to distinguish between intersection and union and effectively explain the differences.

• Use and connect mathematical representations between a Venn diagram and the calculation of the probability of an event by using intersection, union, or complement of the data as appropriate.

• Encourage productive struggle as students explore properties of intersections, unions, and complements and communicate interpretations of these elements for determining the probability of various events. 

Key Understandings

Misconceptions 

• Mutually exclusive events do not share elements of an event and are shown with two non-overlapping circles in a Venn diagram. The probability of mutually exclusive events is said to be zero

• The probability of the intersection (AND) of two independent events is the product of their probabilities.

• A complement (NOT) statement represents the probability of an event not occurring or removing the probability of an event happening, shown as 1 - P(A)

• The probability of the union (OR) of two events equals the sum of the probabilities of each individual event minus the probability of the intersection of the events. 

• Students will interchange the use of the words intersection "and" and union "or" .

• Students may confuse properties of unions and intersections and incorrectly determine when it is more appropriate to add, subtract, or multiply probabilities of events. Students misinterpret the compliment as the element(s) represented by the remaining data values that do not exist in the event. Students may not realize that a null set, represented by a mutually exclusive event, is considered to be an empty set.

• Students may confuse the notations for union ∪ and intersection ⋂. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. (7.D.2.2)

• Calculate experimental probabilities and represent them as percents, fractions, and decimals between 0 and 1. (PA.D.2.1)

• Define, compare, and contrast the probabilities of dependent and independent events. (PA.D.2.3)

• Apply simple counting procedures (factorials, permutations, combinations, and tree diagrams) to determine sample size, sample space, and calculate probabilities. (A1.D.2.1) 

• Apply probability concepts to real-world situations to make informed decisions. (A1.D.2.4)

• Assess the validity of a logical argument and give counterexamples to disprove a statement. (G.RL.1.3)

• Evaluate reports by making inferences, justifying conclusions, and determining appropriateness of data collection methods. Show how graphs and data can be distorted to support different points of view. (A2.D.2.1)

•  Identify and explain misleading conclusions and graphical representations of data sets. (A2.D.2.2)

• Understand that two events, A and B, are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if two events are independent. (S.P.2.1)

• Understand and calculate the conditional probability of A given B as P(A and B)/P(B). (S.P.2.2)

• Interpret independence of A and B as saying that the conditional probability of A, given B, is the same as the probability of A. (S.P.2.3)