In June of 2013, it came to light that the U.S. government had been tapping into the records of the world’s largest internet and telecommunications companies in an attempt to gather information about potential threats to national security. One of these programs, codenamed PRISM, included direct access to the servers of companies like Google, Facebook, and Apple.

In this lesson, students use conditional probabilities to examine some of the implications of a program like PRISM. Specifically, if someone has been identified as a threat, what’s the probability that person actually is a threat?

### Students will

• Describe qualitatively the relationships among subsets of the population, given several Euler Diagrams
• Interpret conditional probabilities from a context and calculate the probabilities of complements
• Calculate the joint probability of two events based on their conditional probabilities
• Apply the Law of Total Probability in order to calculate the probability that someone is dangerous, given that he or she has been identified by PRISM as dangerous

### Before you begin

The main prerequisite for this lesson is a familiarity with the definition of conditional probability. Though not strictly necessary (because it can be deduced from the diagrams), it will also be helpful if students know the relationship between the joint probability of two events and their conditional probabilities: i.e., P (A ∩ B) = P(A | B) ⋅ P(B). The Law of Total Probability is used in the final calculation, but the lesson guides students toward its use in an intuitive way, so they need not be familiar with it up front.