Introduction
Probability is one of the cornerstones of AP Statistics. Whether it’s multiple choice questions on independence, or FRQs involving random variables, probability shows up everywhere. Many students lose points because they forget the basic rules or don’t recognize when to apply them.
This guide will walk you through the must-know probability rules for AP Statistics—complete with explanations, formulas, and examples—so you can walk into the exam with confidence.
1. The Foundation: Probability Basics
Before diving into advanced rules, let’s recap the fundamentals.
- Probability Formula:
P(A)=Number of favorable outcomesTotal number of outcomesP(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} - Range of Probability:
0≤P(A)≤10 \leq P(A) \leq 1- 0 = impossible
- 1 = certain
- 0.5 = equally likely
✅ Example: Rolling a fair die:
- P(even)=36=0.5P(\text{even}) = \frac{3}{6} = 0.5
2. The Addition Rule
The addition rule helps with “OR” probabilities.
- General Rule:
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B) - If A and B are mutually exclusive (can’t both happen), then:
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
✅ Example:
- Rolling a die: P(2 or even)=P(2)+P(even)−P(2∩even)P(\text{2 or even}) = P(2) + P(\text{even}) - P(2 \cap \text{even})
- = 16+36−16=36=0.5\frac{1}{6} + \frac{3}{6} - \frac{1}{6} = \frac{3}{6} = 0.5
3. The Multiplication Rule
The multiplication rule helps with “AND” probabilities.
- General Rule:
P(A∩B)=P(A)×P(B∣A)P(A \cap B) = P(A) \times P(B|A) - If A and B are independent:
P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)
✅ Example:
- Flipping two coins: P(HH)=P(H)×P(H)=0.5×0.5=0.25P(\text{HH}) = P(H) \times P(H) = 0.5 \times 0.5 = 0.25
4. The Complement Rule
Sometimes it’s easier to calculate the opposite event.
- Rule:
P(Ac)=1−P(A)P(A^c) = 1 - P(A)
✅ Example:
- P(at least one head in 2 flips)=1−P(no heads)=1−(0.5×0.5)=0.75P(\text{at least one head in 2 flips}) = 1 - P(\text{no heads}) = 1 - (0.5 \times 0.5) = 0.75
5. Conditional Probability
Conditional probability asks: What’s the chance of event B, given that A already happened?
- Rule:
P(B∣A)=P(A∩B)P(A)P(B|A) = \frac{P(A \cap B)}{P(A)}
✅ Example:
- A bag has 3 red and 2 blue balls. Pick one at random. What’s the probability of blue given the first was red?
- After removing red, there are 4 balls left: 2 blue, 2 red.
- P(blue | red first)=24=0.5P(\text{blue | red first}) = \frac{2}{4} = 0.5
6. Independence vs Dependence
- Independent: Knowing one event happens doesn’t affect the probability of the other.
- Example: Two coin flips.
- Dependent: The outcome of one changes the probability of the other.
- Example: Drawing cards without replacement.
✅ Exam Tip: If the problem says “without replacement,” assume dependence.
7. The Law of Total Probability
If an event can happen in multiple disjoint ways, you sum the probabilities.
- Rule:
P(B)=∑P(B∣Ai)P(Ai)P(B) = \sum P(B|A_i)P(A_i)
✅ Example:
- Suppose 60% of students are girls and 40% are boys. 30% of girls and 20% of boys play sports.
- P(sports)=(0.6)(0.3)+(0.4)(0.2)=0.18+0.08=0.26P(\text{sports}) = (0.6)(0.3) + (0.4)(0.2) = 0.18 + 0.08 = 0.26
8. Bayes’ Theorem
Bayes’ theorem lets you “reverse” conditional probability.
- Formula:
P(A∣B)=P(B∣A)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}
✅ Example:
- A test is 95% accurate. 1% of population has disease.
- P(disease | positive test)=0.95(0.01)(0.95)(0.01)+(0.05)(0.99)P(\text{disease | positive test}) = \frac{0.95(0.01)}{(0.95)(0.01) + (0.05)(0.99)}
- ≈ 0.16 (only 16% chance!)
9. Random Variables and Probability Rules
- Expected Value (Mean):
E(X)=∑x⋅P(x)E(X) = \sum x \cdot P(x) - Variance:
Var(X)=∑(x−μ)2P(x)Var(X) = \sum (x - \mu)^2 P(x) - Rules for Sums of Random Variables:
- E(X+Y)=E(X)+E(Y)E(X+Y) = E(X) + E(Y)
- Var(X+Y)=Var(X)+Var(Y)Var(X+Y) = Var(X) + Var(Y) (if independent)
✅ These appear in AP Stats FRQs frequently.
10. Common Mistakes on the Exam
- ❌ Mixing up “OR” and “AND” rules
- ❌ Forgetting to subtract overlap in addition rule
- ❌ Assuming independence when events are dependent
- ❌ Misinterpreting “at least one” (always use complement rule!)
- ❌ Forgetting to answer in context on FRQs
11. Probability Calculator Tips (TI-84/89)
Your calculator is a lifesaver on test day.
- 2nd → Vars (DISTR menu): Use
binompdf,binomcdf,normalcdf,invNorm - binompdf(n, p, x): Exact probability
- binomcdf(n, p, x): Cumulative probability (≤ x)
- normalcdf(low, high, μ, σ): Normal distribution probability
✅ Practice using these before the exam—they save precious minutes.
12. Final Exam Strategies for Probability
- Use tree diagrams or tables if confused
- Look for keywords:
- “At least one” → complement
- “Without replacement” → dependence
- “Or” → addition rule
- “And” → multiplication rule
- Always write your answers in words on FRQs
Conclusion
Mastering these AP Statistics probability rules will give you a major edge on the exam. They form the foundation of random variables, inference, and simulations that show up in both multiple choice and FRQs.
By drilling these rules and practicing with RevisionDojo’s AP Stats guides, you’ll avoid careless mistakes and maximize your chances of scoring a 5.
Frequently Asked Questions
Q: Do I need to memorize all the probability rules?
A: Yes—these are not on the formula sheet, so you must know them cold.
Q: What’s the biggest probability mistake students make?
A: Forgetting to subtract overlap in the addition rule.
Q: How can I get faster at probability problems?
A: Use your calculator for binomial & normal distributions, and practice recognizing keywords.
Q: Will Bayes’ Theorem be on the exam?
A: Yes, often in FRQs or multiple choice questions about conditional probability.
Q: Is probability harder than inference in AP Stats?
A: Many students struggle with inference more, but probability errors cost lots of points if ignored.
