AP Statistics 2.1-2.2: Estimating Probabilities and Introduction to Probability

Chance experiment

A planned procedure performed under controlled conditions whose result is not predetermined. Flipping a coin or rolling a die is a chance experiment.

Outcome

A single possible result of an experiment, such as heads on one coin flip or rolling a 4.

Sample space

The set of every possible outcome of an experiment, usually denoted by $S$. It can be represented by a list, tree diagram, or Venn diagram.

Event in probability

A collection or subset of outcomes from a sample space, commonly labeled with uppercase letters such as $A$ or $B$.

How is the probability of an event written, and what does it represent?

$P(A)$ denotes the probability of event $A$ and measures the event's likelihood of occurring. It corresponds to the long-run relative frequency expected after many repetitions.

What values can a probability take?

Every probability satisfies $0 \le P(A) \le 1$. A probability of 0 means the event is impossible, while a probability of 1 means it is certain.

What does $P(A)=0.5$ mean?

Event $A$ is equally likely to occur and not occur. This does not mean that exactly half of a small number of trials must produce $A$.

Equally likely outcomes

Outcomes are equally likely when each has the same probability of occurring. For a fair six-sided die, each face has probability $1/6$.

How do you calculate an event's probability when all sample-space outcomes are equally likely?

$P(A)=\dfrac{\text{number of outcomes in }A}{\text{total number of outcomes in }S}$.

A fair dime and a fair nickel are tossed. What is the probability of getting exactly one head?

The equally likely outcomes are $\{HH,HT,TH,TT\}$. Exactly one head occurs in $HT$ and $TH$, so $P(\text{exactly one head})=2/4=1/2$.

A fair six-sided die is rolled. What is the probability of rolling at least 5?

The favorable outcomes are $\{5,6\}$, so $P(\text{at least 5})=2/6=1/3$.

Law of large numbers

As the number of repetitions increases, the observed relative frequency of an event tends to approach its theoretical probability. It does not require the relative frequency to equal the theoretical value in every finite sample.

Why can a short series of fair-die rolls differ substantially from theoretical probabilities?

Random outcomes do not follow a fixed repeating pattern. The law of large numbers predicts closer agreement only over a very large number of trials.

How does a simulation estimate the probability of an event?

Model the chance process, repeat it many times, and estimate the probability with the resulting relative frequency: $\text{estimated probability}=\dfrac{\text{number of trials in which the event occurs}}{\text{total number of trials}}$. More repetitions generally produce a more reliable estimate.

Biased or unfair probability experiment

An experiment is biased when its outcomes are not equally likely, such as a loaded die or an improperly balanced coin. In that case, the favorable-outcome count divided by the total number of outcomes may not give the correct probability.

What is the difference between theoretical probability and empirical probability?

Theoretical probability is based on a model or known assumptions, whereas empirical probability is estimated from observed relative frequencies in actual trials.

What does the event $A$ OR $B$ contain?

$A\text{ OR }B$ contains every outcome in $A$, in $B$, or in both. In set notation it is $A\cup B$, and shared outcomes are counted only once.

What does the event $A$ AND $B$ contain?

$A\text{ AND }B$ contains outcomes that belong to both events simultaneously. In set notation it is $A\cap B$.

What is the complement of an event?

The complement $A'$ consists of all outcomes in the sample space that are not in $A$. It represents the event that $A$ does not occur.

What probability relationship always connects an event and its complement?

$P(A)+P(A')=1$, so equivalently $P(A')=1-P(A)$.

A fair die is rolled and $A$ is the event of rolling an even number. What is $A'$?

$A=\{2,4,6\}$, so $A'=\{1,3,5\}$. Both events have probability $1/2$.

Mutually exclusive events

Events are mutually exclusive if they cannot happen together; they have no outcomes in common. Mathematically, $A\cap B=\varnothing$ and $P(A\cap B)=0$.

What is the general addition rule for an OR event?

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The intersection is subtracted because outcomes shared by both events would otherwise be counted twice.

What addition rule applies to mutually exclusive events?

If $A$ and $B$ are mutually exclusive, then $P(A\cup B)=P(A)+P(B)$ because their intersection has probability zero.

A fair die is rolled. Let $C=\{3,5\}$ and $D=\{2,4\}$. Why are $C$ and $D$ mutually exclusive?

No die outcome is both odd and even, so $C\cap D=\varnothing$ and $P(C\cap D)=0$.

A fair die is rolled. Let $C=\{3,5\}$ and $E=\{1,2,3,4\}$. Are $C$ and $E$ mutually exclusive?

No. They share outcome 3, so $C\cap E=\{3\}$ and $P(C\cap E)=1/6\ne0$.

Two fair coins are flipped. Are the outcomes HT and TH the same outcome?

No. They are distinct ordered outcomes: HT means heads first and tails second, whereas TH means tails first and heads second.

Two fair coins are flipped. What is the probability of at least one tail?

The outcomes with at least one tail are $HT$, $TH$, and $TT$, so the probability is $3/4$. Equivalently, use the complement: $1-P(HH)=1-1/4=3/4$.

A coin is followed by a fair six-sided die. How many ordered outcomes are in the combined sample space?

Use the multiplication principle: $2\times6=12$ ordered outcomes, such as $(H,1)$ and $(T,6)$.

What is the multiplication principle for counting outcomes of sequential experiments?

If one stage has $m$ possible outcomes and a second stage has $n$ possible outcomes for each first-stage result, the combined experiment has $mn$ ordered outcomes.