CCSS.Math.Content.HSS-CP.A.2
The Standard
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 they are independent.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students calculate the chance that two events both happen and compare it with the product of the separate chances. They use that comparison to decide whether one event affects the other.
What Mastery Looks Like
- Given a table, diagram, or stated probabilities, students correctly find P(A), P(B), and P(A ∩ B). They justify independence with an equation and explain the result in context.
Common Misconceptions
- Students often confuse independent events with mutually exclusive events. They may multiply probabilities automatically or declare events independent when values are only close because of rounding.
How to Assess It
- Exit ticket: Roll a fair six-sided die. Are the events “roll an even number” and “roll greater than 3” independent? Show three probabilities.
Lesson moves
Ways to Teach It
Give pairs a coin and die; they run 40 trials, tally heads and even rolls, then compare observed joint and product probabilities.
Ask students to explain why drawing a red card changes the chance of drawing another red card when the first card is not replaced.
Give teams event-pair cards using dice, cards, and spinners; they calculate three probabilities and sort each pair as independent or dependent.
Use a two-way table about school lunch choice and grade level; students test whether the two categories appear independent.
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Printable HSS-CP.A.2 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSS-CP.A.2, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
Download the worksheetLearning progression
Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSS-CP.A.1
Independence requires interpreting A and B occurring together as an intersection of events in the sample space.
- CCSS.Math.Content.7.SP.C.8
Finding compound event probabilities with tables or tree diagrams supports comparing P(A and B) to P(A)P(B) for independence.
- CCSS.Math.Content.7.NS.A.3
Multiplying and comparing rational probabilities supports testing whether joint probability equals the product of individual probabilities.
What This Unlocks
Mastery here sets students up for these next.
- CCSS.Math.Content.HSS-CP.A.3
Students carry forward the product test for independence to connect conditional probabilities matching original probabilities.
- CCSS.Math.Content.HSS-CP.A.5
The product rule gives a precise test for independence that students can translate into everyday explanations of related events.
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