Two-way tables and conditional probability — Digital SAT Math practice
Read conditional probabilities from a table.
Key terms
- Conditional Probability: The likelihood of an event occurring given that another event has already happened, restricting the sample space to the given condition.
- Two-Way Table: A grid displaying frequencies or counts for two categorical variables, with row and column totals summarizing the data.
- Joint Frequency: The count of observations that fall into both a specific row category and a specific column category simultaneously.
- Marginal Total: The sum of frequencies in a single row or column, representing the total count for that specific category regardless of the other variable.
Common traps
- Using the grand total as the denominator instead of the specific group total specified by the given condition.
- Confusing P(A given B) with P(B given A) by swapping the numerator and denominator roles.
- Forgetting to verify that row and column totals match the grand total before calculating probabilities.
Key ideas to know
Identifying the Denominator in Conditional Probability
When asked for P(prefer math | grade 10), which group is the denominator? A) All students B) Grade 10 students C) Math lovers D) Grade 9 students
B. The phrase 'given they are in grade 10' restricts the sample space to only grade 10 students, making them the denominator.
Watch out: Using the grand total or the target group total instead of the 'given' group total for the denominator.
Setting Up the Conditional Probability Fraction
Given a table where 40 students play soccer and 15 of those also swim, what is the correct fraction for P(swim | soccer)? A) 15/60 B) 15/40 C) 40/15 D) 25/40
B. P(A given B) uses group B as the total. Soccer players are the 'given' group (denominator = 40). Swimmers within that group are the target (numerator = 15).
Watch out: Flipping the fraction or using the grand total instead of the 'given' row/column total.
Wrong denominator
A table shows that 30 of 50 seniors and 20 of 40 juniors take calculus. A student says P(calculus | junior) = 20/90 because there are 90 students in all. What is wrong?
The word “given” limits the group to the 40 juniors, so the denominator is 40. P(calculus | junior) = 20/40 = 1/2.
Watch out: Using the grand total instead of the total of the “given” row or column.