Biostatistics: Probability and Screening Tests

This set of vocabulary flashcards covers fundamental concepts of probability, mathematical notations, classical and empirical methods, basic probability rules, independence, conditional probability, and the performance metrics of medical screening tests.

Probability

The mathematical description of randomness and uncertainty, measuring the likelihood of a random phenomenon or chance behavior.

Certain

A term used to describe an event that is definitely going to happen, with a probability of $1$.

Likely

A term used to describe an event that will probably happen but is not definite.

Unlikely

A term used to describe an event that will probably not happen, but might.

Impossible

A term used to describe an event that is definitely not going to happen, with a probability of $0$.

Experiment

A planned operation carried out in a controlled manner.

Chance experiment

An experiment wherein the result is not pre-determined.

Outcome

The result of an experiment.

Sample space

Represented by the symbol $S$, it is the set of all possible outcomes of an experiment.

Event

Any combination of outcomes, typically represented by upper case letters.

Equally likely

A condition where each outcome of an experiment occurs with equal probabilities.

Basic Principles of Probability

Rules stating that for any event $A$, $0 \le P(A) \le 1$, where $0$ is impossible and $1$ is certain.

Unusual event

An event that has a low probability of occurring.

Theoretical (Classical) methods

Method used for games of chance where probabilities are determined by the scenario itself using logic or probability rules.

Empirical (Observational) methods

Method that uses a series of trials producing outcomes that cannot be predicted in advance, making use of relative frequency.

Relative frequency of event A

$$\frac{\text{number of times A occurred}}{\text{total number of repetitions}}$$

Law of large numbers

Characteristic stating that as the number of repetitions of an experiment increases, the relative frequency tends to become closer to the theoretical probability.

Complement Rule

The probability that an event does not occur, calculated as $P(\text{not } A) = 1 - P(A)$, given that $A$ and $not A$ make up all possible outcomes.

Disjoint (Mutually Exclusive) events

Two events that cannot occur at the same time, meaning $P(A \text{ and } B) = 0$.

Addition Rule for Disjoint Events

If $A$ and $B$ are disjoint, then $P(A \text{ or } B) = P(A) + P(B)$.

General Addition Rule

The rule for events that are not disjoint: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$, used to avoid double-counting the overlap.

Independent Events

Events where the fact that one event has occurred does not affect the probability that the other event will occur.

Dependent Events

Events where the occurrence of one event does affect the probability that the other event will occur.

Multiplication Rule for Independent Events

If $A$ and $B$ are independent, then $P(A \text{ and } B) = P(A) \times P(B)$.

Conditional Probability

The probability of event $B$ occurring given that event $A$ has already occurred, denoted as $P(B | A) = \frac{P(A \text{ and } B)}{P(A)}$.

Relative risk (RR)

A ratio of the probability of an event occurring in the exposed group versus the non-exposed group, expressed as $RR = \frac{P(B | A)}{P(B | \text{not } A)}$.

Total probability rule

A theorem used to find the probability of an event by breaking it into distinct parts based on related events: $P(A) = P(A \text{ and } B) + P(A \text{ and } \text{not } B)$.

Gold screening test

In medicine, the diagnostic test or benchmark that is regarded as the definitive or final solution for determining disease status.

True Positive

The number of individuals with the disease who receive a positive screening test result.

True Negative

The number of individuals without the disease who receive a negative screening test result.

False Positive

The number of individuals without the disease who receive a positive screening test result.

False Negative

The number of individuals with the disease who receive a negative screening test result.

Sensitivity

The ability of a test to correctly identify those with the disease; expressed as $P(\text{symptom}+ | \text{disease}+)$, or true positives divided by total with disease.

Specificity

The ability of a test to correctly identify those free of disease; expressed as $P(\text{symptom}- | \text{disease}-)$, or true negatives divided by total without disease.

Positive Predictive Value (PV+)

The probability that a person has the disease given that they tested positive ($P(\text{disease} | \text{test}+)$).

Negative Predictive Value (PV-)

The probability that a person does not have the disease given that they tested negative ($P(\text{no disease} | \text{test}-)$).

Bayes' Theorem

A rule stating that the conditional probability of an event, based on the occurrence of another event, is equal to the likelihood of the second event given the first multiplied by the probability of the first event.