# Using conditional probability to interpret data

Revisit two-way frequency tables and use them to approximate conditional probabilities (S-CP.A.4).

In the One Variable Statistics unit, students encountered two-way frequency tables. In this section, students revisit these tables, using them to approximate conditional probabilities by using the tables as a sample space.

## Tasks

WHAT: Students are presented with a two-way frequency table that shows passengers on the Titanic categorized in two ways: by survival and by ticket class. Students are asked to calculate probabilities using the data from the table. The task affords students the opportunity to attend to precision (MP.6) as they answer the questions and compare the subtle differences in the wording of the questions and the large differences in their answers.

WHY: The emphasis is on developing students’ understanding of conditional probability. Students should understand the difference in meaning between the probability of A and B, the probability of A given B, and the probability of B given A.

## External Resources

#### Description

WHAT: The teacher’s guide gives this description: “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?”

WHY: PRISN provides a different lens for understanding conditional probabilities. Instead of frequency tables, different possibilities are presented in Euler diagrams (similar to those in the IM task on the Addition Rule in section 2) and probabilities are presented as percentages. Students make use of the fact that the probability of an event and the probability of its complement have sum 1.

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#### Description

WHAT: The teacher’s guide gives this description: “In 1973, the graduate school at the University of California, Berkeley, admitted 44% of its male applicants, but only 35% of its female applicants. As a result, the university was later sued over gender discrimination in its admissions process. In court, however, the university pointed out that most of the graduate departments actually admitted women at a higher rate than their male peers. How could that be? In this lesson, students use frequency tables and conditional probability to explore Simpson’s Paradox and try to settle the discrimination case once and for all.”

WHY: Golden Gatekeepers offers a context for interpreting two-way frequency tables (S-ID.B.5) and to think about conditional probabilities (S-CP.A.4). Students are asked to analyze the claims made by the plaintiffs in the case and explain how organizing the data in a different way could lead to a different conclusion (MP.3).

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