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AP Statistics Unit 2 Visual ReviewA topic-by-topic visual walkthrough of Unit 2: Probability, Random Variables, and Probability Distributions — two-way tables, probability rules, random variables, the binomial and normal distributions, and the Central Limit Theorem.
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TOPIC 2.1
Two Categorical Variables: Tables
TWO-WAY TABLE
Yes No Total
Male 30 20 50
Female 45 15 60
Total 75 35 110
Marginal & joint
MARGINAL = a row/column total's share.
JOINT = one cell ÷ grand total.
P(Male & Yes) = 30/110 ≈ 0.27
Show with a segmented / mosaic bar chart.
Each variable's categories partition the whole.
A two-way table shows a relationship
Rows = one variable, columns = another. Row/column totals are the "margins" — hence marginal distributions.
A two-way table gives marginal (totals) and joint (cells) distributions.
The Review Hub · AP Statistics Unit 2
TOPIC 2.2
Two Categorical Variables: Summaries
Conditional distributions
A CONDITIONAL distribution restricts to
ONE row or column, then finds percents.
P(Yes | Male) = 30/50 = 0.60
Divide the cell by that group's TOTAL,
not the grand total.
Association
Two categorical variables are ASSOCIATED
if the conditional distributions DIFFER
across groups.
If P(Yes|Male) ≠ P(Yes|Female), there's
an association.
Same percentages → no association.
Compare conditional percentages
To judge association from a table, always compare CONDITIONAL percentages, not raw counts (group sizes differ).
Conditional distributions differ → association ; divide by the group total.The Review Hub · AP Statistics Unit 2
TOPIC 2.3
Estimating Probability by Simulation
Imitate chance with trials
A SIMULATION uses random digits, coins,
or a calculator to imitate a random process.
P(event) ≈ (# successes) / (# trials)
Repeat many times and take the relative
frequency as the estimate.
Steps to set up a simulation
1. Assign digits to outcomes.
2. Define one trial & a "success."
3. Run many trials; count successes.
4. Estimate = successes / trials.
More trials → estimate approaches true probability.
The Law of Large Numbers
As the number of trials grows, the observed proportion of successes converges to the true probability.
Estimate a probability by simulation: successes ÷ trials over many trials.
The Review Hub · AP Statistics Unit 2
TOPIC 2.4
Introduction to Probability
// basic probability rules
0 ≤ P(A) ≤ 1 P(sample space) = 1
complement: P(Aᶜ) = 1 − P(A)
equally likely: P(A) = (# favorable) / (# total)
Probability is long-run relative frequency
The probability of an event is the proportion of times it would occur over MANY repetitions.
roll a die: P(even) = 3/6 = 0.5
The complement rule is often the fastest route to an answer.
0 ≤ P(A) ≤ 1, P(Aᶜ) = 1 − P(A) — probability = long-run relative frequency.The Review Hub · AP Statistics Unit 2
TOPIC 2.5
Mutually Exclusive Events
general addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually exclusive (disjoint)
Two events that CAN'T both happen:
P(A ∩ B) = 0
Then the addition rule simplifies:
P(A ∪ B) = P(A) + P(B)
e.g. rolling a 2 and rolling a 5 on one die.
Worked example
Draw a card: P(King or Queen)?
A card can't be both → disjoint.
= 4/52 + 4/52 = 8/52 ≈ 0.15
The ∪ ("or") means "at least one of them."
Disjoint ≠ independent — they're different ideas.
P(A∪B)=P(A)+P(B)−P(A∩B) ; disjoint means P(A∩B)=0.The Review Hub · AP Statistics Unit 2
TOPIC 2.6
Conditional Probability
P(A | B) = P(A ∩ B) / P(B)
"Given that B happened"
P(A | B) is the probability of A once we
KNOW B occurred — restrict to B's world.
From a table, divide by B's total.
P(Yes|Male) = 30/50 = 0.60
The condition becomes the new denominator.
Multiplication rule
P(A ∩ B) = P(B)·P(A | B)
Rearranging the conditional formula gives
the AND (∩) probability.
Use a TREE DIAGRAM to organize
sequential events.
P(A|B) = P(A∩B)/P(B) — the condition is the new denominator.The Review Hub · AP Statistics Unit 2
TOPIC 2.7
Independent Events
independent ⟺ P(A|B) = P(A) ⟺ P(A∩B) = P(A)·P(B)
One doesn't affect the other
Events are INDEPENDENT if knowing B
doesn't change the probability of A.
P(A∩B∩C) = P(A)P(B)P(C)
Multiply for independent events in a row.
Coin flips are independent; cards without replacement aren't.
Disjoint ≠ independent
Mutually exclusive events (P(A∩B)=0) are
actually DEPENDENT:
if A happens, B definitely can't →
knowing A changes P(B).
Don't confuse the two — a classic exam trap.
Independent: P(A∩B)=P(A)·P(B) ; disjoint events are NOT independent.
The Review Hub · AP Statistics Unit 2
TOPIC 2.8
Random Variables & Distributions
A random variable X
assigns a NUMBER to each outcome of a
random process.
Its PROBABILITY DISTRIBUTION lists each
value with its probability.
All the probabilities must sum to 1.
// X = number of heads in 2 flips
x 0 1 2
P(x) .25 .50 .25
Sum = .25 + .50 + .25 = 1 ✓
DISCRETE: countable values (table).
CONTINUOUS: area under a density curve.
Probability = area for continuous X
For a continuous random variable, probabilities are AREAS under a density curve; the total area equals 1.
A random variable's probabilities sum to 1 ; continuous → area under a curve.
The Review Hub · AP Statistics Unit 2
TOPIC 2.9
Parameters of Random Variables
// mean (expected value) and variance of a discrete X
μₓ = E(X) = Σ xᵢ · P(xᵢ)
σₓ² = Σ (xᵢ − μₓ)² · P(xᵢ) σₓ = √(variance)
Combining random variables
E(aX+b) = a·E(X) + b
Var(aX+b) = a²·Var(X)
Means add: E(X±Y)=E(X)±E(Y).
Adding a constant b doesn't change the variance.
Variances add if INDEPENDENT
Var(X+Y) = Var(X) + Var(Y)
Var(X−Y) = Var(X) + Var(Y)
Variances ADD even for a DIFFERENCE.
Add variances, never standard deviations.
μₓ=Σx·P(x) ; means add, and independent variances add (even for X−Y).The Review Hub · AP Statistics Unit 2
TOPIC 2.10
The Binomial Distribution
// n independent trials, each success prob p
P(X = k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ
μ = np σ = √(np(1−p))
The BINS conditions
• Binary: success/failure each trial
• Independent trials
• Number of trials n is FIXED
• Same probability p every trial
Worked example
10 free throws, p = 0.7; P(exactly 8)?
= C(10,8)(.7)⁸(.3)² ≈ 0.233
μ = 10(.7) = 7 made on average.
C(n,k) counts the orderings of k successes.
Binomial: P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, μ=np, σ=√(np(1−p)) .
The Review Hub · AP Statistics Unit 2
TOPIC 2.11
The Normal Distribution
μ−2σ μ−σ μ μ+σ μ+2σ
68% within 1σ · 95% within 2σ · 99.7% within 3σ
z-scores standardize any value
z = (x − μ) / σ → look up the area (proportion) below x in the normal table
Normal: 68–95–99.7 rule; z = (x−μ)/σ gives the proportion below x.
The Review Hub · AP Statistics Unit 2
TOPIC 2.12
Sampling Distributions & the CLT
// sampling distribution of the sample mean x̄
mean of x̄ = μ SD of x̄ = σ / √n
for proportions: mean p̂ = p, SD = √(p(1−p)/n)
Central Limit Theorem
For a large sample (n ≥ 30), the sampling
distribution of x̄ is APPROXIMATELY NORMAL —
no matter the population's shape.
This is what makes inference (Units 3–4) possible.
Bigger n → less variability
The SD of x̄ (σ/√n) SHRINKS as n grows —
larger samples give more precise estimates.
x̄ is an UNBIASED estimator of μ (centered
at the true mean).
CLT : for large n, x̄ is ~Normal with mean μ and SD σ/√n.The Review Hub · AP Statistics Unit 2
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How to use the visual review
Spend 30 seconds per slide before clicking next. Look at the diagram, then ask yourself: "Could I state this probability rule, or the mean and SD of this distribution, from memory?"
Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.
This is great for review the night before the exam — fast, visual, and covers every rule and model you need to recognize in Unit 2.