Connecting Benchmarks/Horizontal Alignment

Terms from the K-12 Glossary

• Conditional relative Frequency 
• Event 
• Experimental probability 
• Frequency table 
• Joint frequency 
• Sample space 
• Theoretical probability

Vertical Alignment

Previous Benchmarks

• MA.7.DP.2.2
• MA.7.DP.2.3
• MA.7.DP.2.4 
• MA.8.DP.2.2
• MA.8.DP.2.3

Next Benchmarks

Purpose and Instructional Strategies

• Independence in this sense means that knowing whether one event occurred does not change the probability of the other event occurring. For this benchmark, students determine independence using conditional probabilities. Two events, A and B, are independent if P(A | B) = P(A) and P(B | A) = P(B). 
• Students also use the product of probabilities to check independence (MA.912.DP.4.2). 
• If P(A ∩ B) = P(A) × P(B), then the events are independent. 
• Be sure to distinguish independence from mutually exclusive events. 
  • In mutually exclusive events (A ∩ B) = 0. This means that the two events cannot occur at the same time. 
  • Instruction includes the understanding that P(A | B) ≠ P(B | A) unless P(A) = P(B).
  • When we check for independence in real-world data sets, it's rare to get perfectly equal probabilities. We often assume that events are independent and test that assumption on sample data. If the probabilities are significantly different, then we conclude the events are not independent.

Common Misconceptions or Errors

• Students may confuse what it means to be dependent and independent. 
• Students may confuse independence with mutually exclusive events. 
• Students may have difficulty recalling how to convert fractions, decimals and percentages. 
• Students may think the symbol used for conditional probability is a slash that would be used to represent division and simply divide the probability of A by the probability of B. 
• Students may get confused as to which event probability should be the denominator. 
• Students may get confused when working with a two-way table that they need to restrict their answer to a certain section that is from the “given” conditional piece. 
  • For example, when given the condition of male they are only looking in the row or column containing males to get the total. 
• Students who have difficulty with the terminology and notation will also have difficulty in understanding what is being asked by the questions.

Instructional Tasks

• The table below displays the results of a survey of eating preferences.
  
      

• Part A. Name two events that can be represented by the two-way table. 
• Part B. Are the two events in Part A independent events? Explain why or why not using conditional probability.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.