Introduction to Probability and Statistics

Vocabulary practice flashcards covering fundamental probability concepts, expected value, tree diagrams, compound probabilities (OR and AND problems), and conditional probability.

Experiment

A controlled operation that yields a set of results.

Outcomes

The possible results of an experiment.

Event

A sub collection of the outcomes of an experiment.

Sample Space

The collection or list of all possible outcomes of an experiment.

Empirical Probability (Experimental Probability)

The relative frequency of occurrence of an event determined by actual observations of an experiment, calculated as: $P(E) = \frac{\text{number of times event } E \text{ has occurred}}{\text{total number of times the experiment has been performed}}$

Theoretical Probability

Probability determined through a study of possible outcomes that can occur for a given experiment; if a procedure has $n$ different simple events with an equal chance of occurring and $s$ is the number of ways event $E$ can occur, then $P(E) = \frac{s}{n}$.

Subjective Probability

An educated guess of probability found by simply guessing or estimating its value based on knowledge of the relevant circumstances.

Expected Value (Expectation)

The expected results of an experiment or business venture over the long run, given by the formula $E = P_1 A_1 + P_2 A_2 + P_3 A_3 + \dots + P_n A_n$, where $P_i$ is the probability of event $i$ and $A_i$ is the net amount won or lost.

Fair Price

The cost to play that results in a net expected value of zero, calculated as $\text{Fair Price} = \text{Expectation} + \text{Cost to play}$.

Fair Game

A game or venture in which the expected value (expectation) is $0$, meaning that over the long run a player neither gains nor loses points or money.

Counting Principle

A principle stating that if a first experiment can be performed in $M$ distinct ways and a second experiment can be performed in $N$ distinct ways, the two experiments in that order can be performed in $M \times N$ distinct ways.

Sample Point (Simple Event)

Each individual outcome in the sample space of an experiment.

Without Replacement

A selection process where once an item is selected, it is not returned to the pool and it is impossible to select that same item again.

Tree Diagram

A branching diagrammatic representation that systematically lists all the sample points in the sample space.

Compound Probabilities (OR Problems)

Probabilities requiring a successful outcome for at least one of the given events (either $A$ or $B$ or both $A$ and $B$), calculated using the addition formula: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

Mutually Exclusive Events

Two events that cannot occur simultaneously ($P(A \text{ and } B) = 0$), simplifying the addition formula to $P(A \text{ or } B) = P(A) + P(B)$.

Independent Events

Events where the occurrence of either event in no way affects the probability of occurrence of the other, typically modeled by sampling with replacement.

Dependent Events

Events where the occurrence of one event affects the probability of occurrence of the other, typically modeled by sampling without replacement.

Conditional Probability

The probability that event $B$ occurs given that event $A$ has happened, represented by $P(B \mid A)$ and read as 'probability of $B$ given $A$'.

Incidence Rate

The rate or proportion at which a specific condition or disease naturally occurs within a defined population (e.g., $0.6\%$ for breast cancer among symptom-free women aged $40$ to $50$).

False Negative Rate

The probability that a diagnostic screening test incorrectly indicates a negative result for an individual who actually has the condition or virus.

False Positive Rate

The probability that a diagnostic screening test incorrectly indicates a positive result for an individual who does not have the condition or virus.