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Flashcards · AP® Statistics · Unit 2

Unit 2: Probability, Random Variables, and Probability Distributions flashcards

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2.1 Tabular and Graphical Representations for the Distributions of Two Categorical Variables

What is a two-way table?
A table showing two categorical variables for the same individuals: rows are categories of one variable, columns of the other. Each cell counts individuals in that row and column; totals sit at the edges.
How do you compare groups in a two-way table?
Find the distribution of the response within each group: cell ÷ group total. Group A 40 of 50 Yes (0.80) is more likely than group B 60 of 100 (0.60), even with fewer Yes answers.
When are two categorical variables associated?
When the distribution of one variable changes across the categories of the other. If both bars of a segmented bar chart divide at 0.45, there is no association.
What does a segmented bar chart show?
One bar of height 1 per group, split into within-group proportions, so you can see at a glance whether distributions differ. It does not show group sizes.
How do you read a mosaic plot?
Bar widths show group sizes and segment heights show within-group proportions. Cell count = width × height × grand total: 0.30 × 0.50 × 600 = 90.
Which variable goes on the axis of a segmented or side-by-side bar chart?
The explanatory variable (one bar or cluster per group), split by the response variable. Use proportions if group sizes differ.
Can a two-way table from a survey show cause and effect?
No. A survey is observational, so an association does not show cause.

2.2 Summary Statistics for Two Categorical Variables

What is a joint relative frequency?
One cell as a share of all individuals ("A and B"): cell count ÷ grand total. 30 of 100 people in Group A who said Yes gives 0.30.
What is a marginal relative frequency?
A row or column total as a share of all individuals, describing one variable on its own: 40 Yes of 100 gives 0.40.
What is a conditional relative frequency?
A cell as a share of one row or column total ("of the A, what proportion are B"). Of Group A, 30 ÷ 50 = 0.60 said Yes; of the Yes answers, 30 ÷ 40 = 0.75 came from Group A. Direction matters.
How are joint, marginal and conditional relative frequencies linked?
Joint = marginal × conditional, for example 0.40 × 0.50 = 0.20.
How do you judge association from relative frequencies?
Compare conditional proportions within each explanatory group, not counts or joint values. If the conditional distributions are all equal (and equal to the marginal), there is no association.
What is a conditional distribution?
The set of conditional relative frequencies within one group; it adds to 1.

2.3 Estimating Probabilities Using Simulation

What is probability in terms of relative frequency?
The relative frequency of an event over a very large number of trials of a random process. A relative frequency from real or simulated data estimates it; 0.2 means about 1 in 5 in the long run, not exactly.
What is the law of large numbers?
With independent trials, the relative frequency of an event settles toward one value, the probability, as the number of trials increases.
How do you assign random digits for a simulation?
Use single digits for probabilities like 0.1, 0.2 and pairs 00 to 99 for 0.35 or 0.07. For probability 0.45, pairs 00 to 44 = event. For three equally likely outcomes, assign 1 to 3 and ignore the rest.
What are the steps of a simulation?
State the event and outcome probabilities; assign random values (say which to ignore and whether to skip repeats); describe one trial and what you record; do many trials, recording count and total; estimate and interpret in context.
What is the gambler's fallacy?
Believing an outcome is "due" after a run without it. Independent trials have no memory: after 5 reds on a fair spinner, the probabilities for the next spin are unchanged.
How do you estimate a probability from a simulation?
Trials with the event ÷ total trials: 92 of 400 gives 0.23. It is an estimate; another run gives a different value.

2.4 Introduction to Probability

What is a sample space?
The set S of all possible outcomes, with no overlaps and none missing. P(S) = 1, and the probabilities of all separate outcomes add to 1.
How do you find a probability with equally likely outcomes?
P(E) = outcomes in E ÷ outcomes in S. A fair spinner numbered 1 to 5: P(even) = 2/5 = 0.4. Only valid when outcomes are equally likely.
What is the complement rule?
P(Eᶜ) = 1 − P(E), where Eᶜ means E does not happen. P(E) = 0.27 gives P(Eᶜ) = 0.73. The complement of "at least 2" is "at most 1".
How do you find P(at least one)?
P(at least one) = 1 − P(none). Two fair coins: P(no heads) = 1/4, so P(at least one head) = 3/4.
How can you tell a probability model is invalid?
Any probability below 0 or above 1, or probabilities of all outcomes not adding to 1. P(red) = 0.45, P(blue) = 0.35, P(green) = 0.3 adds to 1.1, so it is invalid.
How do you list outcomes for a two-stage process?
Use a grid or systematic list of ordered pairs. Count the equally likely pairs, not the totals or scores they produce.

2.5 Mutually Exclusive Events

What is the intersection of two events?
A ∩ B, the outcomes in both A and B ("A and B"). The joint probability P(A ∩ B) is the probability both happen on the same trial; from a two-way table it is cell count ÷ grand total.
What does mutually exclusive mean?
The events cannot happen at the same time on the same trial, so P(A ∩ B) = 0. Any joint probability above 0, however small, means they are not mutually exclusive.
How do you justify whether events are mutually exclusive?
Show P(A ∩ B) and compare with 0. Cards 1 to 20: "multiple of 9" and "even and greater than 15" share 18, so P(both) = 0.05, above 0: not mutually exclusive.
Are mutually exclusive events the same as complements?
No. An event and its complement are always mutually exclusive, but disjoint events can leave outcomes in neither event.
Does a zero cell in a two-way table prove a combination is impossible?
No. It shows P(A ∩ B) = 0 for an individual chosen from that data set only; the combination may simply not appear in the sample.
Is "mutually exclusive" the same as "independent"?
No. Mutually exclusive means cannot happen together; independence means one does not affect the probability of the other.

2.6 Conditional Probability

What is conditional probability?
P(A | B), the probability of A given that B has happened: P(A | B) = P(A ∩ B) / P(B), with P(B) above 0. P(A ∩ B) = 0.12 and P(B) = 0.4 give 0.3.
How do you find a conditional probability from a two-way table?
Divide the cell for A and B by the total for the condition B, not by the grand total. Words like "of the", "among", "given that" name the condition.
What is the general multiplication rule?
P(A ∩ B) = P(A) · P(B | A), finding a joint probability in stages. P(A) = 0.5 and P(B | A) = 0.3 give 0.15.
How do you handle drawing without replacement?
The second draw is conditional: reduce the totals after the first. 6 working and 2 broken bulbs, two taken: P(both broken) = 2/8 × 1/7 = 1/28 ≈ 0.0357.
How do you use a tree diagram?
Later branches are conditional on the earlier path. Branches from one point add to 1; multiply along a path; add the paths that make up an event.
How do you reverse a condition?
P(B | A) = P(A ∩ B) / P(A): find the joint probability first, then divide by the new condition. Keep exact values until the end. P(late | van) is not P(van | late).

2.7 Independent Events and Unions of Events

What does it mean for two events to be independent?
Knowing whether A happened does not change the probability of B. Check with one of: P(A | B) = P(A), P(B | A) = P(B), or P(A ∩ B) = P(A) · P(B), showing both numbers.
How do you check independence numerically?
P(A) = 0.2, P(B) = 0.6, P(A ∩ B) = 0.15: P(A) · P(B) = 0.12, and 0.12 ≠ 0.15, so not independent. Compare P(A | B) with P(A), not with P(B).
What is the general addition rule?
P(A ∪ B) = P(A) + P(B) − P(A ∩ B); the overlap is subtracted once because it is counted in both. "Or" includes "both". For mutually exclusive events, P(A ∪ B) = P(A) + P(B).
When can you use P(A ∩ B) = P(A) · P(B)?
Only for independent events, when independence is given, checked or justified by how outcomes arise (separate spins). Independent A and B with 0.5 and 0.3 give 0.15, and P(A ∪ B) = 0.65.
How do you find P(at least one) for repeated independent trials?
1 − P(none) = 1 − (1 − p)ⁿ, not n × p. Three alarms each failing with probability 0.1: 1 − 0.9³ = 0.271.
Are mutually exclusive events independent?
No. Disjoint events with non-zero probabilities are dependent: if one happens, the other's probability becomes 0.

2.8 Introduction to Random Variables and Probability Distributions

What is a random variable?
A variable (capital X) that gives a numerical value to each outcome of a random process; small x is one particular value. Discrete random variables have values you can list.
What makes a probability distribution valid?
Every probability is between 0 and 1 and they add to 1. Values 0, 1, 2, 3 with 0.2, 0.35, k, 0.15 give k = 1 − 0.70 = 0.3. Check both conditions, not only the sum.
What is a cumulative probability distribution?
It gives P(X ≤ x) for each x. P(X = x) = P(X ≤ x) − P(X ≤ previous value): with P(X ≤ 1) = 0.3 and P(X ≤ 2) = 0.65, P(X = 2) = 0.35 and P(X above 2) = 0.35.
How do you translate "at least" and "between" with a cumulative table?
P(X ≥ a) = 1 − P(X ≤ value just below a). P(X above a and at most b) = P(X ≤ b) − P(X ≤ a). "Fewer than 3" excludes 3.
How can a probability distribution be found or shown?
Exactly from probability rules (state their assumptions, often independence), or estimated by simulation with relative frequencies. Show it as a table, probability histogram or function, one row per value of X.
How do you find an unknown constant in a probability function?
Set the sum of all probabilities equal to 1 and solve. P(X = x) = x ÷ 6 for x = 1, 2, 3 is valid: 1/6 + 2/6 + 3/6 = 1.

2.9 Parameters of Random Variables

How do you calculate the expected value of a discrete random variable?
μ = E(X) = Σ xᵢ · P(xᵢ), the probability-weighted average. It is the long-run average value, not what will happen on one trial, and need not be a possible value.
How do you calculate the standard deviation of a random variable?
σ² = Σ (xᵢ − μ)² · P(xᵢ), then σ = √σ². No n − 1: probabilities do the weighting. Y = 10 (0.2) or 20 (0.8): μ = 18, σ² = 64(0.2) + 4(0.8) = 16, σ = 4 minutes.
What does the standard deviation of a random variable mean?
The typical distance of the values of X from its mean μ over the long run.
Is a simulated mean a parameter or a statistic?
A statistic: it changes from sample to sample. μ, from the probability distribution, is the parameter. A mean gain of $0.35 from 200 simulated plays is a statistic.
Which calculator value gives σ for a probability distribution?
Use the probability list as frequencies and read σx, not Sx.
What is a quick check on a calculated mean?
Σ (xᵢ − μ) · P(xᵢ) = 0: the deviations balance at the mean.

2.10 The Binomial Distribution

What are the conditions for a binomial random variable?
Each trial has two outcomes; the number of trials n is fixed; trials are independent; the probability of success p is the same on every trial. State each in context. Then X ~ B(n, p), counting successes.
What is the binomial probability formula?
P(X = x) = ₙCₓ · pˣ · (1 − p)ⁿ⁻ˣ, where ₙCₓ = n! ÷ [x!(n − x)!]. X ~ B(4, 0.5): P(X = 2) = 6 × 0.0625 = 0.375. Leaving out ₙCₓ gives only one order.
What are the mean and standard deviation of a binomial distribution?
μ = np and σ = √[np(1 − p)]. B(50, 0.2): μ = 10, σ = √8 ≈ 2.83. Do not report the variance np(1 − p) as σ.
How do you translate words to binomial inequalities?
"At least 3": X ≥ 3. "More than 3": X ≥ 4. "At most 3": X ≤ 3. "Fewer than 3": X ≤ 2. P(X ≥ x) = 1 − P(X ≤ x − 1).
How do you find P(X ≥ 1) for a binomial variable?
1 − P(X = 0) = 1 − (1 − p)ⁿ. B(6, 0.1): 1 − 0.9⁶ = 0.4686.
When is a count not binomial?
When n is not fixed, or when draws are without replacement from a small group so p changes. 3 cards drawn from 10 with 4 red: not binomial.
How should a binomial calculation be written up?
Name the distribution, n, p and the values, not just calculator syntax, and interpret the result in context.

2.11 The Normal Distribution

What are the properties of a normal distribution?
Continuous, unimodal, symmetric and bell-shaped, fixed by μ (centre) and σ (spread). Smaller σ: taller and more concentrated. The standard normal is N(0, 1).
What is the empirical rule?
For a normal distribution, about 68%, 95% and 99.7% of values lie within 1, 2 and 3 σ of μ. One-side strips: 34%, 13.5%, 2.35%, 0.15%. X ~ N(40, 5): about 2.5% above 50.
What is the probability of a single exact value of a continuous random variable?
Zero. Only intervals have probabilities.
How do you find a normal probability?
Standardise with z = (x − μ) / σ (divide by σ, not σ²), then find the area. X ~ N(40, 5): P(X below 33) has z = −1.4, area 0.0808. A z-table gives area to the left.
How do you find the value that cuts off the top p%?
The area to the left is 1 − p/100; find z, then x = μ + zσ. Top 20% of N(40, 5): z = 0.8416, x = 44.21.
How do you handle "middle p%" or "most extreme p%" questions?
Middle p%: (1 − p/100)/2 in each tail. Most extreme p%: half of p/100 in each tail, not all in one.
What should a normal calculation show?
Four things: the distribution and parameters, the boundary, the direction (sketch), and the answer. Check the model fits: roughly unimodal and symmetric, and values sensible (can they go below 0?).

2.12 Sampling Distributions and the Central Limit Theorem

What is a sampling distribution?
The distribution of a statistic (such as x̄ or p̂) over all possible samples of the same size from the same population. It is not the distribution of one sample's data.
What is sampling variability?
A statistic varies from sample to sample.
What does the central limit theorem say?
The sampling distribution of a sample mean is approximately normal, and the approximation improves as n increases, even for a skewed population. It does not make the population or the sample normal.
What happens to a sampling distribution as n increases?
The centre stays near the parameter, variability shrinks and the shape of x̄ gets closer to normal. More repetitions of a simulation only give a clearer picture; they do not shrink the spread.
What is a randomization distribution?
From an experiment: keep the responses and group sizes, shuffle the treatment labels many times and recompute the statistic each time.
How do you judge whether an observed result is unusual from a simulation?
Find the proportion of simulated values as extreme as or more extreme than the observed one: 12 of 400 reallocations (3%) is unusual. It is evidence against the assumption, not proof.
How do you simulate a sampling distribution?
Assume a parameter value (and say so), take a random sample of size n, record the statistic, and repeat many times.

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