Probability, Random Variables, and Probability Distributions

What this unit covers

The topics below follow the published Statistics course framework for Unit 2. This unit is worth 15–25% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Scatterplots & Association13 min · 3 objectivesIdentify explanatory and response variables and plot them correctly · Describe a scatterplot by direction, form, strength, and unusual features · Distinguish association from causation
• Correlation13 min · 3 objectivesInterpret the correlation coefficient r as a measure of linear association · State the properties and limitations of correlation · Explain why a strong correlation does not confirm a linear relationship
• Least-Squares Regression15 min · 3 objectivesInterpret the slope and y-intercept of a least-squares regression line in context · Use a regression equation to make predictions and recognize extrapolation · Interpret the coefficient of determination r²
• Residuals & Model Fit13 min · 3 objectivesCompute and interpret a residual in context · Use a residual plot to assess whether a linear model is appropriate · Explain the danger of extrapolation and the effect of influential points
• Outliers, Leverage & Influential Points in Regression14 min · 3 objectivesDistinguish a regression outlier from a high-leverage point · Predict how each affects the slope, intercept and correlation · Explain why removing a point requires justification rather than convenience

Formulas in Unit 2

Every term in Unit 2

All 70 terms we publish for Probability, Random Variables, and Probability Distributions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Central Limit Theorem

For large n, the sampling distribution of x̄ is approximately Normal regardless of population shape (n ≥ 30 rule of thumb).

Combining random variables

Means always add: μ(X ± Y) = μX ± μY. Variances add only when X and Y are independent, and they add even for a difference.

Sampling distribution of x̄

Center μ, standard deviation σ/√n. Normal if the population is normal, or approximately normal for large n by the Central Limit Theorem.

Sampling distribution of p̂

Center p, standard deviation √(p(1 − p)/n), approximately normal when np ≥ 10 and n(1 − p) ≥ 10.

Law of large numbers

As the number of trials grows, the observed proportion approaches the true probability. It says nothing about short runs.

Complement rule

P(not A) = 1 − P(A). Often the fastest route to "at least one" problems.

Addition rule

P(A or B) = P(A) + P(B) − P(A and B). The subtraction avoids double-counting the overlap.

Mutually exclusive vs independent

Mutually exclusive events cannot both occur; independent events do not affect each other's probability. Mutually exclusive events with non-zero probability are never independent.

Conditional probability

P(A | B) = P(A and B)/P(B). The condition restricts the sample space to B.

Checking independence

A and B are independent when P(A | B) = P(A). Verify numerically rather than assuming.

Expected value

E(X) = Σx·P(x), the long-run average over many repetitions. It need not be an attainable value.

Variance of a random variable

Var(X) = Σ(x − μ)²P(x). Standard deviation is its square root, interpreted as typical distance from the expected value.

Binomial setting

Binary outcomes, Independent trials, fixed Number of trials, and constant probability of Success — BINS.

Binomial mean and standard deviation

μ = np and σ = √(np(1 − p)).

Geometric setting

Same as binomial but counting trials until the first success, so the number of trials is not fixed. Mean is 1/p.

10% condition

When sampling without replacement, trials are close enough to independent provided the sample is no more than 10% of the population.

Sampling distribution

The distribution of a statistic over all possible samples of a given size. Not the distribution of one sample, and not the population.

Unbiased estimator

A statistic whose sampling distribution is centered at the parameter. The sample mean and sample proportion are both unbiased.

Standard error

The standard deviation of a sampling distribution. It shrinks as √n, so quadrupling the sample size halves it.

Why sample size matters more than population size

Standard error depends on n, not on population size, which is why a national poll needs about the same sample as a state poll.

Difference of two proportions

Center p₁ − p₂ with standard deviation √(p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂) — the variances add even though the means subtract.

Simulation design

State the model, how one trial is run, what counts as a success, and how many trials. Then report the estimated probability as a proportion.

"At least one" problems

Compute the complement — the probability of none — and subtract from 1. Far faster than summing every case.

Two-way table probabilities

Joint probability uses the grand total as the denominator; conditional probability uses the row or column total.

Tree diagrams

Multiply along a branch for a joint probability, then add across branches that satisfy the condition.

Binomial vs geometric on the exam

A fixed number of trials counting successes is binomial; counting trials until the first success is geometric.

Linear transformation of a random variable

For Y = a + bX, the mean transforms the same way but the standard deviation is multiplied by |b| and the addition has no effect on spread.

Independence of two random variables

Required before adding variances. Without it, the variance of a sum is not the sum of the variances.

Distinguishing three distributions

The population, the distribution of one sample, and the sampling distribution of a statistic are different objects. Exam questions turn on telling them apart.

Shape of the sampling distribution of p̂

Approximately normal when np and n(1 − p) are both at least 10. Otherwise it is skewed and normal-based methods do not apply.

Why standard error shrinks with n

It divides by √n, so precision improves with sample size but with diminishing returns — quadrupling n only halves the error.

Bias vs variability

Bias is being centered in the wrong place; variability is being spread out. A method can have low variability and still be badly biased.

Sample space

The set of all possible outcomes of a chance process. Listing it correctly is usually most of the work in a probability problem, and a tree or table is the systematic way to do it.

General multiplication rule

P(A and B) = P(A)P(B | A), valid always. It reduces to P(A)P(B) only when the events are independent, which is why the conditional form is the one to remember.

Multiplication rule for independent events

P(A and B) = P(A)P(B), valid only under independence. Applying it to dependent events — drawing without replacement is the standard case — is a common error.

Disjoint events

Events that cannot both occur, so P(A and B) = 0 and P(A or B) = P(A) + P(B). Disjoint is not independent; disjoint events with nonzero probability are strongly dependent.

Marginal versus joint probability

A marginal probability concerns one variable ignoring the other and comes from a row or column total. A joint probability concerns both simultaneously and comes from a single cell.

Conditional probability from a two-way table

Restrict attention to the row or column given, then divide the cell by that row or column total. The given event determines the denominator, which is the whole content of conditioning.

Reversing a conditional

P(A | B) and P(B | A) are different quantities and are related through the overall rates of A and B. A tree diagram or a table of natural frequencies makes the reversal concrete without any formula.

Designing a simulation

State how digits or random numbers map to outcomes, what constitutes one trial, what to ignore, how many trials, and what you record. Vague mappings are the usual reason a simulation description loses credit.

Discrete versus continuous random variables

Discrete variables take countable values and have a probability for each; continuous variables take any value in an interval, so probability is area under a density curve and any single value has probability zero.

Requirements of a probability distribution

Every probability lies between 0 and 1 and the total is exactly 1. Checking this is a free first step whenever a table of probabilities is given.

Expected value is a long-run average

It is what the mean of the outcomes approaches over many repetitions, and it need not be an attainable value — an expected family size of 2.3 children is a perfectly sensible answer.

Variance of a linear transformation

For Y = aX + b, the variance is a²·Var(X) and the standard deviation is |a|·SD(X). The additive constant b shifts the distribution and never affects its spread.

Mean of a sum

μ(X + Y) = μX + μY, always, whether or not X and Y are independent. Expected value is the one operation that requires no conditions.

Variance of a difference

σ²(X − Y) = σ²X + σ²Y for independent variables — variances ADD even for a difference, because subtracting a random quantity makes the result more variable, not less.

Binomial conditions (BINS)

Binary outcomes, Independent trials, a fixed Number of trials, and the same Success probability throughout. Failing any one of the four rules the binomial model out.

Binomial probability formula

P(X = k) = C(n, k)p^k(1 − p)^(n−k). The combination counts the arrangements, and the powers give the probability of any one specific arrangement.

Cumulative binomial probabilities

Questions phrased "at most", "at least" or "fewer than" require summing several terms. Translate the words into an inequality first, and watch whether the endpoint is included.

Geometric setting versus binomial

Same conditions as binomial except that the number of trials is not fixed — you count trials until the first success. The mean number of trials is 1/p.

Geometric probability formula

P(X = k) = (1 − p)^(k−1)p: k − 1 failures followed by a success. Note the exponent is one less than k, which is the usual slip.

Solving at-least-one problems by complement

Compute the complement: P(at least one) = 1 − P(none). Summing the individual cases is almost always slower and more error-prone.

Law of large numbers and the gambler's fallacy

The proportion of successes approaches the true probability as trials accumulate. It says nothing about short runs, which is why the gambler's fallacy — expecting a correction after a streak — is wrong.

Expected value is not the most likely value

For a binomial with n = 5 and p = 0.3 the expected value is 1.5, which cannot occur. Expected value is a balance point, not a mode.

Independence of random variables

Knowing the value of one gives no information about the other. It is the condition for adding variances, and it is a claim to be justified from the context rather than assumed by default.

Parameter and statistic notation

Parameters describe populations and use Greek letters or capitals: μ, σ, p. Statistics describe samples: x̄, s, p̂. Using the wrong symbol in a hypothesis statement costs the point.

Center of the sampling distribution of p̂

μ(p̂) = p. The sample proportion is an unbiased estimator, so over many samples it neither systematically overshoots nor undershoots.

Spread of the sampling distribution of p̂

σ(p̂) = √(p(1 − p)/n), provided the sample is no more than 10% of the population. It shrinks with √n, so quadrupling the sample halves the spread.

Center of the sampling distribution of x̄

μ(x̄) = μ, whatever the sample size and whatever the shape of the population. Unbiasedness of the sample mean does not depend on Normality.

Spread of the sampling distribution of x̄

σ(x̄) = σ/√n, again requiring the 10% condition. Note the population standard deviation σ does not change with n; only the sampling distribution narrows.

Large counts condition

np ≥ 10 and n(1 − p) ≥ 10 for the sampling distribution of p̂ to be approximately Normal. It fails for rare events even when n is large, which is when exact binomial methods are needed.

The 10% condition in sampling-distribution formulas

Sampling without replacement makes draws slightly dependent. Keeping n below 10% of the population makes that dependence negligible so the standard deviation formulas remain usable.

Sampling distribution of a difference of means

Centered at μ₁ − μ₂ with standard deviation √(σ₁²/n₁ + σ₂²/n₂). Variances add, as always for independent quantities, even though the parameter is a difference.

Why we standardize a statistic

Converting to (statistic − parameter)/standard error puts every procedure on one scale, so a single Normal or t table serves every test. The structure of every test statistic is the same.

Simulating a sampling distribution

Repeatedly draw samples of size n, compute the statistic each time, and plot the results. The plot approximates the theoretical sampling distribution and is how the concept is best seen.

Effect of n on shape

Larger samples make the sampling distribution of x̄ more Normal, regardless of population shape. The more skewed the population, the larger n must be before the approximation is adequate.

Effect of n on center

None. The sampling distribution of an unbiased statistic is centered at the parameter for every sample size — n controls precision, not accuracy.

Variability of the sample median

The sample median is also an unbiased estimator of a symmetric population's center, but its sampling distribution is more variable than the mean's. This is why the mean is preferred when the population is Normal.

Unbiased does not mean correct

An unbiased statistic is right on average over many samples. Any single sample can still be far off, which is exactly why inference reports an interval rather than a number.

Why sample size matters and population size does not

The formulas contain n and not N. Provided the population is much larger than the sample, a poll of 1,000 is about as precise for a country as for a town.

What examiners penalize here

• Slope interpretations need three parts: the words "predicted" or "on average," the units of y, and the units of x. "For each additional *hour*, the *predicted score* increases by 6 *points*" scores; "the score goes up 6" does not.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 5 units of AP Statistics

1. Unit 1 · Exploring One-Variable Data and Collecting Data
2. Unit 2 · Probability, Random Variables, and Probability Distributions
3. Unit 3 · Inference for Categorical Data: Proportions
4. Unit 4 · Inference for Quantitative Data: Means
5. Unit 5 · Regression Analysis