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Credit Sprint / Study guides / Statistics Unit 2

Credit Sprint's Guide to AP® Statistics, Unit 2: Probability, Random Variables, and Probability Distributions

AP® weight 15–25%

Unit 2 builds the language of chance. You use two-way tables to study the relationship between two categorical variables, apply probability rules including conditional probability and independence, run simulations, and work with discrete and continuous random variables, including the binomial and normal models.

The normal curve and the empirical rule -3σ-2σ-1σmean+1σ+2σ+3σ 68% within 1σ · 95% within 2σ · 99.7% within 3σ

What you need to know

  • A two-way table shows counts for two categorical variables. Joint, marginal and conditional distributions answer different questions. A segmented bar graph or mosaic plot compares conditional relative frequencies. If the conditional distributions are about equal across groups, the variables show little association.
  • Probability is a long-run relative frequency, between 0 and 1. Complement: P(not A) = 1 − P(A). Addition rule: P(A or B) = P(A) + P(B) − P(A and B). Mutually exclusive events cannot happen together, so P(A and B) = 0.
  • Conditional probability: P(A | B) = P(A and B) ÷ P(B). Multiplication rule: P(A and B) = P(A) · P(B | A). Events A and B are independent if P(A | B) = P(A), equivalently P(A and B) = P(A) · P(B).
  • Mutually exclusive and independent are different ideas. Two mutually exclusive events with positive probabilities are never independent, because knowing one happened tells you the other did not.
  • A simulation estimates probability by repeating a random process many times. Describe the model, the trial, what counts as a success and how many trials; more trials give a more stable estimate.
  • A random variable assigns a number to each outcome of a chance process. A discrete random variable has a countable list of values with probabilities that sum to 1. Its mean (expected value) is Σx·P(x) and its standard deviation measures the typical spread around that mean.
  • A linear transformation Y = a + bX changes the mean to a + b·mean and the standard deviation to |b|·SD. Adding a constant does not change the spread.
  • A binomial setting has a fixed number n of trials, each trial is a success or failure, trials are independent, and the probability of success p is constant. Then P(X = k) = C(n,k) p^k (1 − p)^(n−k), with mean np and standard deviation √[np(1 − p)].
  • A continuous random variable is described by a density curve, where area represents probability, and the probability of any single exact value is 0. The normal distribution is the most important example; use z-scores and technology (normalcdf, invNorm) to find areas and percentiles.
  • For a normal model, the percentile of a value is the area to its left. To find a value for a given percentile, work backward with invNorm or a z-table, then convert: x = mean + z · SD.

Exam traps

Watch for'Or' means add and subtract the overlap; 'and' means multiply, but only multiply plain probabilities if the events are independent. Otherwise use P(A) · P(B | A).
Watch forNot independent does not mean mutually exclusive, and mutually exclusive events (with nonzero probability) are never independent.
Watch forThe binomial needs a fixed number of trials. If you stop at the first success, it is not binomial.
Watch forP(X ≥ 3) is not the same as P(X > 3) for a discrete variable. Write out which values are included.
Watch forFor a normal model, always sketch and shade before you compute. Area to the left is the percentile.

Key terms

  • Two-way table: A table of counts for two categorical variables.
  • Marginal distribution: The distribution of one variable alone, found in the margins of a two-way table.
  • Joint relative frequency: The proportion of all individuals that fall in a particular combination of the two variables.
  • Conditional distribution: The distribution of one variable among only those individuals who have a given value of the other.
  • Association between categorical variables: Present when the conditional distributions of one variable differ across the values of the other.
  • Probability: A number from 0 to 1 that tells the long-run relative frequency of an outcome.
  • Law of large numbers: As the number of trials increases, the relative frequency of an outcome gets closer to its true probability.
  • Sample space: The list of all possible outcomes of a chance process.
  • Complement rule: P(not A) = 1 − P(A).
  • Mutually exclusive events: Events that cannot occur together, so P(A and B) = 0.
  • Addition rule: P(A or B) = P(A) + P(B) − P(A and B).
  • Conditional probability: P(A | B) = P(A and B) ÷ P(B), the probability of A given that B occurred.
  • Multiplication rule: P(A and B) = P(A) · P(B | A).
  • Independent events: Events for which knowing that one happened does not change the probability of the other.
  • Tree diagram: A diagram that shows sequential events and their conditional probabilities along branches.
  • Simulation: Using random digits or technology to imitate a chance process many times and estimate a probability.

Practice questions

Five of the 98 Unit 2 questions in Credit Sprint. Every choice is explained in the app.

1. A fair coin is flipped five times and lands heads every time. What is the probability that the sixth flip is heads?
  1. 0.5
  2. Less than 0.5 because tails is due
  3. Greater than 0.5 because heads is due to continue
  4. Cannot be determined
Show answer

0.5. Coin flips are independent, so past results do not change the probability of the next flip.

2. A medical test detects a disease in 95% of people who have it and gives a false positive for 8% of people who do not. If 2% of a population has the disease, which expression gives P(positive)?
  1. 0.02(0.95) + 0.98(0.08)
  2. 0.95 + 0.08
  3. 0.02(0.95)
  4. 0.98(0.92)
Show answer

0.02(0.95) + 0.98(0.08). A positive result can come from a person with the disease (0.02 × 0.95) or a healthy person (0.98 × 0.08); add the two branches.

3. A multiple-choice quiz has 10 questions, each with 4 choices. A student guesses randomly on all of them. Which expression gives the probability of getting exactly 3 correct?
  1. (0.25)³
  2. 3(0.25)
  3. C(10,3)(0.25)³(0.75)⁷
  4. C(10,3)(0.75)³(0.25)⁷
Show answer

C(10,3)(0.25)³(0.75)⁷. The number correct is binomial with n = 10 and p = 0.25, so P(X = 3) = C(10,3)(0.25)³(0.75)⁷.

4. A density curve for a continuous random variable has which property?
  1. The curve can go below the horizontal axis.
  2. The curve must be symmetric.
  3. The total area under the curve is 1.
  4. The height of the curve at a value is the probability of that value.
Show answer

The total area under the curve is 1.. Area under the curve represents probability, and the total probability is 1.

5. To simulate flipping a fair coin, a student uses random digits and lets 0–4 represent heads and 5–9 represent tails. Why is this a valid assignment?
  1. The digits are all different
  2. Each outcome has probability 0.5
  3. Random digits cannot model coins
  4. Heads is lower so it comes first
Show answer

Each outcome has probability 0.5. Five of the ten digits represent each outcome, so heads and tails are equally likely.