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Probability, Random Variables, and Probability Distributions

AP Statistics · Topic 4

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Probability, Random Variables, and Probability Distributions

A basketball player makes five shots in a row. The crowd is sure she is hot tonight. But watch what chance alone can do. Flip a fair coin twenty times. There…

English narration · English + 中文 subtitles burned in

4.1

Random and Non-Random Patterns
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-1
Given that variation may be random or not, conclusions are uncertain.

VAR-1.F
Identify questions suggested by patterns in data. [Skill 1.A]

  • VAR-1.F.1 Patterns in data do not necessarily mean that variation is not random.

Source: College Board AP Course and Exam Description

Something is random 随机 if individual outcomes are uncertain but a regular pattern emerges over many repetitions. Short-run results look erratic; long-run relative frequencies settle down. This long-run stability is what makes probability useful.

Vocabulary Train
English
random/ˈrændəm/
4.2

Estimating Probabilities Using Simulation
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-2
Simulation allows us to anticipate patterns in data.

UNC-2.A
Estimate probabilities using simulation. [Skill 3.A]

  • UNC-2.A.1 A random process generates results that are determined by chance.
  • UNC-2.A.2 An outcome is the result of a trial of a random process.
  • UNC-2.A.3 An event is a collection of outcomes.
  • UNC-2.A.4 Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. All possible outcomes are associated with a value to be determined by chance. Record the counts of simulated outcomes and the count total.
  • UNC-2.A.5 The relative frequency of an outcome or event in simulated or empirical data can be used to estimate the probability of that outcome or event.
  • UNC-2.A.6 The law of large numbers states that simulated (empirical) probabilities tend to get closer to the true probability as the number of trials increases.
    • Illustrative examples for UNC-2.A:
      • An outcome: Rolling a particular value on a six-sided number cube is one of six possible outcomes.
      • An event: When rolling two six-sided number cubes, an event would be a sum of seven. The corresponding collection of outcomes would be $(1, 6)$, $(2, 5)$, $(3, 4)$, $(4, 3)$, $(5, 2)$, and $(6, 1)$, where the ordered pairs indicate (face value on one cube, face value on the other cube).

Source: College Board AP Course and Exam Description

A simulation 模拟 imitates a chance process using random digits or technology. Steps: state the model, assign digits to outcomes, run many trials, and record the proportion of trials meeting the condition. The resulting proportion estimates the probability – more trials give a better estimate.

Vocabulary Train
English
simulation/ˌsɪmjʊˈleɪʃn/
4.3

Introduction to Probability
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.A
Calculate probabilities for events and their complements. [Skill 3.A]

  • VAR-4.A.1 The sample space of a random process is the set of all possible non-overlapping outcomes.
  • VAR-4.A.2 If all outcomes in the sample space are equally likely, then the probability an event E will occur is defined as the fraction: $\dfrac{\text{number of outcomes in event E}}{\text{total number of outcomes in sample space}}$
  • VAR-4.A.3 The probability of an event is a number between 0 and 1, inclusive.
  • VAR-4.A.4 The probability of the complement of an event E, $E'$ or $E^{C}$, (i.e., not E) is equal to $1 - P(E)$.

VAR-4.B
Interpret probabilities for events. [Skill 4.B]

  • VAR-4.B.1 Probabilities of events in repeatable situations can be interpreted as the relative frequency with which the event will occur in the long run.

Source: College Board AP Course and Exam Description

The probability 概率 of an event is a number from $0$ to $1$ giving its long-run relative frequency. The sample space 样本空间 is the set of all outcomes. For an event $A$, the complement 补 rule: $P(A^c)=1-P(A)$. Probabilities of all outcomes sum to $1$.

Probability runs from 0 (impossible) to 1 (certain)
Probability runs from 0 (impossible) to 1 (certain)
The four aces from a deck of playing cards
A deck of cards is a classic source of probability: 52 equally likely outcomes make the chances easy to count
Explore

Explore probability with dice

Probability is the long-run fraction of times an outcome happens. Roll the dice many times and watch the experimental proportions settle toward the theoretical values.

Vocabulary Train
English
probability/ˌprɒbəˈbɪlɪti/
sample space/ˈsæmpl speɪs/
complement/ˈkɒmplɪmənt/
4.4

Mutually Exclusive Events
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.C
Explain why two events are (or are not) mutually exclusive. [Skill 4.B]

  • VAR-4.C.1 The probability that events $A$ and $B$ both will occur, sometimes called the joint probability, is the probability of the intersection of $A$ and $B$, denoted $P(A \cap B)$.
  • VAR-4.C.2 Two events are mutually exclusive or disjoint if they cannot occur at the same time. So $P(A \cap B) = 0$.

Source: College Board AP Course and Exam Description

Two events are mutually exclusive 互斥 (disjoint) if they cannot both happen. Then the addition rule simplifies:

$$P(A\text{ or }B)=P(A)+P(B)\quad(\text{if mutually exclusive}).$$
In general, $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$ – subtract the overlap so it is not counted twice.

A Venn diagram: the overlap is the intersection of two events
A Venn diagram: the overlap is the intersection of two events
Vocabulary Train
English
mutually exclusive/ˈmjuːtʃuːəli eksˈkluːsɪv/
4.5

Conditional Probability
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.D
Calculate conditional probabilities. [Skill 3.A]

  • VAR-4.D.1 The probability that event $A$ will occur given that event $B$ has occurred is called a conditional probability and denoted $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$.
  • VAR-4.D.2 The multiplication rule states that the probability that events $A$ and $B$ both will occur is equal to the probability that event $A$ will occur multiplied by the probability that event $B$ will occur, given that $A$ has occurred. This is denoted $P(A \cap B) = P(A) \cdot P(B \mid A)$.

Source: College Board AP Course and Exam Description

Conditional probability

The conditional probability 条件概率 of $A$ given $B$ is

$$P(A\mid B)=\frac{P(A\text{ and }B)}{P(B)}.$$
It is the chance of $A$ once you know $B$ happened. Two-way tables make these easy: restrict to the row/column for $B$, then find $A$'s share.

On a tree diagram, multiply the probabilities along the branches
On a tree diagram, multiply the probabilities along the branches
Explore

Update a probability on new information

Conditional probability $P(B\mid A)$ is the chance of $B$ once you know $A$ happened. Change the branch probabilities and watch how conditioning reshapes the outcome.

Vocabulary Train
English
conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/
4.6

Independent Events and Unions of Events
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.E
Calculate probabilities for independent events and for the union of two events. [Skill 3.A]

  • VAR-4.E.1 Events $A$ and $B$ are independent if, and only if, knowing whether event $A$ has occurred (or will occur) does not change the probability that event $B$ will occur.
  • VAR-4.E.2 If, and only if, events $A$ and $B$ are independent, then $P(A \mid B) = P(A)$, $P(B \mid A) = P(B)$, and $P(A \cap B) = P(A) \cdot P(B)$.
  • VAR-4.E.3 The probability that event $A$ or event $B$ (or both) will occur is the probability of the union of $A$ and $B$, denoted $P(A \cup B)$.
  • VAR-4.E.4 The addition rule states that the probability that event $A$ or event $B$ or both will occur is equal to the probability that event $A$ will occur plus the probability that event $B$ will occur minus the probability that both events $A$ and $B$ will occur. This is denoted $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Source: College Board AP Course and Exam Description

Events are independent 独立 if knowing one does not change the other's probability: $P(A\mid B)=P(A)$. Then the multiplication rule simplifies:

$$P(A\text{ and }B)=P(A)\,P(B)\quad(\text{if independent}).$$
Independent is not the same as mutually exclusive – mutually exclusive events with nonzero probability are actually dependent (if one happens, the other cannot).

A sample space diagram lists every equally likely outcome
A sample space diagram lists every equally likely outcome
Explore

Combine events with a Venn diagram

For a union $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ — you subtract the overlap so it isn't counted twice. Switch the operation to see each region light up.

Vocabulary Train
English
independent/ˌɪndɪˈpendənt/
4.7

Random Variables and Probability Distributions
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-5
Probability distributions may be used to model variation in populations.

VAR-5.A
Represent the probability distribution for a discrete random variable. [Skill 2.B]

  • VAR-5.A.1 The values of a random variable are the numerical outcomes of random behavior.
  • VAR-5.A.2 A discrete random variable is a variable that can only take a countable number of values. Each value has a probability associated with it. The sum of the probabilities over all of the possible values must be 1.
  • VAR-5.A.3 A probability distribution can be represented as a graph, table, or function showing the probabilities associated with values of a random variable.
  • VAR-5.A.4 A cumulative probability distribution can be represented as a table or function showing the probability of being less than or equal to each value of the random variable.
    • Illustrative examples for VAR-5.A: Outcomes of trials of a random process:
      • The sum of the outcomes for rolling two dice
      • The number of puppies in a randomly selected litter for a certain breed of dog

VAR-5.B
Interpret a probability distribution. [Skill 4.B]

  • VAR-5.B.1 An interpretation of a probability distribution provides information about the shape, center, and spread of a population and allows one to make conclusions about the population of interest.

Source: College Board AP Course and Exam Description

A random variable 随机变量 assigns a number to each outcome of a chance process. A probability distribution 概率分布 lists each possible value with its probability (they sum to $1$). A distribution can be discrete (a table of values) or continuous (an area-under-a-curve model like the normal).

Vocabulary Train
English
random variable/ˈrændəm ˈveərɪəbl/
probability distribution/ˌprɒbəˈbɪlɪti ˌdɪstrɪˈbjuːʃn/
4.8

Mean and Standard Deviation of Random Variables
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-5
Probability distributions may be used to model variation in populations.

VAR-5.C
Calculate parameters for a discrete random variable. [Skill 3.B]

  • VAR-5.C.1 A numerical value measuring a characteristic of a population or the distribution of a random variable is known as a parameter, which is a single, fixed value.
  • VAR-5.C.2 The mean, or expected value, for a discrete random variable $X$ is $\mu_X = \sum x_i \cdot P(x_i)$.
  • VAR-5.C.3 The standard deviation for a discrete random variable $X$ is $\sigma_X = \sqrt{\sum (x_i - \mu_x)^2 \cdot P(x_i)}$.

VAR-5.D
Interpret parameters for a discrete random variable. [Skill 4.B]

  • VAR-5.D.1 Parameters for a discrete random variable should be interpreted using appropriate units and within the context of a specific population.

Source: College Board AP Course and Exam Description

The mean (expected value) 期望值 of a discrete random variable is the probability-weighted average:

$$\mu_X=E(X)=\sum x_i\,P(x_i).$$
The standard deviation $\sigma_X=\sqrt{\sum (x_i-\mu_X)^2\,P(x_i)}$ measures typical spread from the mean. The expected value is the long-run average outcome, not a value you expect on any single trial.

Worked example. A game pays $\$5$ with probability $0.2$ and costs you $\$1$ (a $-1$ outcome) with probability $0.8$. The expected value is

$$E(X)=5(0.2)+(-1)(0.8)=1-0.8=\$0.20,$$
so over many plays you gain about $20$ cents per play on average, even though no single play gives exactly that.

Vocabulary Train
English
mean (expected value)/miːn/
4.9

Combining Random Variables
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-5
Probability distributions may be used to model variation in populations.

VAR-5.E
Calculate parameters for linear combinations of random variables. [Skill 3.B]

  • VAR-5.E.1 For random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$.
  • VAR-5.E.2 Two random variables are independent if knowing information about one of them does not change the probability distribution of the other.
  • VAR-5.E.3 For independent random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$, and the variance of $aX + bY$ is $a^2\sigma^2_x + b^2\sigma^2_y$.

VAR-5.F
Describe the effects of linear transformations of parameters of random variables. [Skill 3.C]

  • VAR-5.F.1 For $Y = a + bX$, the probability distribution of the transformed random variable, $Y$, has the same shape as the probability distribution for $X$, so long as $a > 0$ and $b > 0$. The mean of $Y$ is $\mu_y = a + b\mu_x$. The standard deviation of $Y$ is $\sigma_y = |b|\sigma_x$.

Source: College Board AP Course and Exam Description

When you add or subtract random variables, means add: $\mu_{X\pm Y}=\mu_X\pm\mu_Y$. If $X$ and $Y$ are independent, variances add (even when subtracting):

$$\sigma^2_{X\pm Y}=\sigma^2_X+\sigma^2_Y.$$
Take the square root for the standard deviation. Also, scaling: $\mu_{aX+b}=a\mu_X+b$ and $\sigma_{aX+b}=|a|\sigma_X$.

4.10

Introduction to the Binomial Distribution
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.A
Estimate probabilities of binomial random variables using data from a simulation. [Skill 3.A]

  • UNC-3.A.1 A probability distribution can be constructed using the rules of probability or estimated with a simulation using random number generators.
  • UNC-3.A.2 A binomial random variable, $X$, counts the number of successes in $n$ repeated independent trials, each trial having two possible outcomes (success or failure), with the probability of success $p$ and the probability of failure $1 - p$.

UNC-3.B
Calculate probabilities for a binomial distribution. [Skill 3.A]

  • UNC-3.B.1 The probability that a binomial random variable, $X$, has exactly $x$ successes for $n$ independent trials, when the probability of success is $p$, is calculated as $P(X = x) = \binom{n}{x} p^x (1 - p)^{n-x}, x = 0, 1, 2, \ldots, n$. This is the binomial probability function.

Source: College Board AP Course and Exam Description

The binomial distribution

A binomial 二项 setting (BINS): a fixed number $n$ of Independent trials, each with two outcomes (success/failure) and the same success probability $p$. The random variable $X=$ number of successes. Its probability:

$$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}.$$

The binomial distribution, with mean n times p
The binomial distribution, with mean n times p
Explore

Shape a binomial distribution

A binomial distribution counts successes in $n$ independent trials each with probability $p$. Change $n$ and $p$ and watch the bars shift and spread.

Vocabulary Train
English
binomial/baɪˈnəʊmɪəl/
4.11

Parameters for a Binomial Distribution
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.C
Calculate parameters for a binomial distribution. [Skill 3.B]

  • UNC-3.C.1 If a random variable is binomial, its mean, $\mu_x$, is $np$ and its standard deviation, $\sigma_x$, is $\sqrt{np(1 - p)}$.

UNC-3.D
Interpret probabilities and parameters for a binomial distribution. [Skill 4.B]

  • UNC-3.D.1 Probabilities and parameters for a binomial distribution should be interpreted using appropriate units and within the context of a specific population or situation.

Source: College Board AP Course and Exam Description

For a binomial $X$ with $n$ trials and success probability $p$:

$$\mu_X=np,\qquad \sigma_X=\sqrt{np(1-p)}.$$
Use these for "how many successes do we expect, and how much do they vary" questions.

Worked example. A player makes $70\%$ of free throws. In $n=10$ shots, the probability of exactly $8$ makes is

$$P(X=8)=\binom{10}{8}(0.7)^8(0.3)^2=45\times0.0576\times0.09\approx0.23,$$
and the expected number of makes is $\mu=np=10(0.7)=7$, with $\sigma=\sqrt{10(0.7)(0.3)}\approx1.45$.

4.12

The Geometric Distribution
Syllabus
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.E
Calculate probabilities for geometric random variables. [Skill 3.A]

  • UNC-3.E.1 For a sequence of independent trials, a geometric random variable, $X$, gives the number of the trial on which the first success occurs. Each trial has two possible outcomes (success or failure) with the probability of success $p$ and the probability of failure $1 - p$.
  • UNC-3.E.2 The probability that the first success for repeated independent trials with probability of success $p$ occurs on trial $x$ is calculated as $P(X = x) = (1 - p)^{x-1} p, x = 1, 2, 3, \ldots$. This is the geometric probability function.

UNC-3.F
Calculate parameters of a geometric distribution. [Skill 3.B]

  • UNC-3.F.1 If a random variable is geometric, its mean, $\mu_x$, is $\dfrac{1}{p}$ and its standard deviation, $\sigma_x$, is $\dfrac{\sqrt{(1 - p)}}{p}$.

UNC-3.G
Interpret probabilities and parameters for a geometric distribution. [Skill 4.B]

  • UNC-3.G.1 Probabilities and parameters for a geometric distribution should be interpreted using appropriate units and within the context of a specific population or situation.

Source: College Board AP Course and Exam Description

A geometric 几何 setting is the same as binomial but with no fixed $n$: you keep trying until the first success. The random variable $Y=$ the trial of the first success:

$$P(Y=k)=(1-p)^{k-1}\,p,\qquad \mu_Y=\frac{1}{p}.$$
So the expected number of trials until the first success is $1/p$.

Vocabulary Train
English
geometric/ˌdʒiːəʊˈmetrɪk/
4.12

Exam tips
  • A probability lies in $[0,1]$; use the complement ($1-P$) and add mutually exclusive events.
  • For independent events multiply; for "and/or" use the general addition and conditional rules.
  • Expected value = $\sum(\text{value}\times\text{probability})$.
  • Recognise binomial (fixed $n$, two outcomes, constant $p$) and geometric settings.
  • Draw a tree or table for multi-stage problems and multiply along branches.

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