AP Statistics · Unit 2 · Topic 2.7

Independent Events and Unions Study Lab

Two events are independent when knowing one happened does not change the probability of the other. The general addition rule finds the probability that at least one of two events happens. Set the probabilities, watch the Venn diagram, and check independence three ways.

1. Check independence, then add

Move the sliders or choose a preset. The joint probability is limited by the other two, so the sliders keep it possible.

Independent ⇔ P(A ∩ B) = P(A) · P(B) P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Preset situations

Probabilities

Venn diagram probability in each region

P(A or B)
P(A | B)
P(A) · P(B)
Independent?

Three ways to test independence

General addition rule

Repeated independent trials: "at least one"

For independent events you can multiply. The fastest route to "at least one" is the complement: P(at least one) = 1 − P(none) = 1 − (1 − p)ⁿ

Show your work

2. Key ideas

Independence

Knowing one tells you nothing

A and B are independent when P(A | B) = P(A). Equivalently P(B | A) = P(B), or P(A ∩ B) = P(A) · P(B). If any one of these holds, all three do.

Multiplication rule

"And" for independent events

Only when A and B are independent can you find P(A and B) by multiplying P(A) · P(B). In general, P(A and B) = P(A) · P(B | A).

Union

"Or" means at least one

P(A ∪ B) is the probability that A happens, B happens, or both. Adding P(A) + P(B) counts the overlap twice, so subtract P(A ∩ B) once.

Not the same

Independent ≠ mutually exclusive

Mutually exclusive events cannot happen together, so P(A ∩ B) = 0. If both have positive probability, knowing A happened tells you B did not. That makes them dependent.

Complement

At least one

"At least one success in n independent trials" has many cases. Its complement, "no successes," has only one: P(none) = (1 − p)ⁿ. Subtract from 1.

From data

Independence in a two-way table

With sample data, compare conditional relative frequencies. If they are close, the variables look independent in the sample. Exact equality is rare in real data.

3. Quick reference

The rules for "and" and "or," and when each shortcut is allowed.

QuestionRuleWhen you can use itExample
P(A or B)P(A) + P(B) − P(A ∩ B)Always0.5 + 0.4 − 0.2 = 0.7
P(A or B), shortcutP(A) + P(B)Only if A and B are mutually exclusiveP(roll 1 or 2) = 1/6 + 1/6
P(A and B)P(A) · P(B | A)Always0.5 × 0.4 = 0.2
P(A and B), shortcutP(A) · P(B)Only if A and B are independentTwo coin flips: 0.5 × 0.5 = 0.25
Independence checkIs P(A | B) = P(A)?Compare the numbers; do not assume0.2/0.4 = 0.5 = P(A), so independent
At least one in n trials1 − (1 − p)ⁿTrials independent, same p1 − 0.9⁵ = 0.410

4. On the FRQ

How to earn the points

  • To show independence, compute and compare: "P(A | B) = 0.20 ÷ 0.40 = 0.50, which equals P(A) = 0.50, so A and B are independent."
  • To show dependence, show the two numbers differ: "P(late | rain) = 0.30 but P(late) = 0.12. Since these are not equal, the events are not independent."
  • Write the rule before plugging in: "P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = …"
  • When you multiply probabilities, say why it is allowed: "Because the trials are independent, P(all 3) = 0.9³."
  • Answer in context with a probability between 0 and 1, rounded to at least 3 decimal places if it is not exact.

Common mistakes

  • Multiplying P(A) · P(B) for "and" when the events are not known to be independent.
  • Adding P(A) + P(B) for "or" when the events overlap, which can give an answer above 1.
  • Thinking mutually exclusive means independent. They are opposites when both probabilities are positive.
  • Computing "at least one" by multiplying p by n. That is not a probability rule and can exceed 1.
  • Declaring sample data "independent" or "dependent" without comparing specific relative frequencies.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 2.7. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.