Premed · Premed · Statistics Biostatistics
Lecture 6: Probability Fundamentals
Statistics / Biostatistics
Learning Objectives
By the end of this lecture, students will be able to:
- Define probability and describe the three main interpretations (classical, frequentist, subjective)
- Apply the basic rules of probability (addition, complement, multiplication)
- Distinguish between mutually exclusive and independent events
- Use sample spaces, event notation, and Venn diagrams to solve probability problems
- Apply probability concepts to clinical scenarios
Lecture Content
I. What Is Probability?
Probability quantifies the likelihood of an event occurring and is expressed as a number between 0 (impossible) and 1 (certain). There are three main interpretations. The classical (theoretical) interpretation is based on equally likely outcomes, where P(event) = number of favorable outcomes / total number of possible outcomes. For example, the probability of rolling a 3 on a fair die is 1/6. The frequentist (empirical) interpretation is based on long-run relative frequency, where P(event) equals the number of times the event occurs divided by the number of trials as the trials approach infinity. If a drug cures 800 out of 1000 patients, the estimated P(cure) is 0.80. The subjective (Bayesian) interpretation reflects a personal degree of belief based on available information. A clinician might estimate a 70% chance that a patient has pneumonia based on symptoms and physical examination.
II. Basic Definitions
An experiment (trial) is a process that produces an observable result, such as administering a treatment and observing recovery. The sample space (S) is the set of all possible outcomes; for a coin flip, S = {Heads, Tails}. An event is a subset of the sample space. A simple event consists of a single outcome, while a compound event consists of two or more outcomes. The complement of event A (denoted A^c or A') includes all outcomes in S that are not in A, and its probability is P(A^c) = 1 - P(A).
III. Rules of Probability
Three fundamental rules govern all probability calculations. Rule 1 states that for any event A, the probability must satisfy 0 <= P(A) <= 1. Rule 2 states that P(S) = 1, meaning the probability of the entire sample space is 1. Rule 3 is the complement rule: P(A^c) = 1 - P(A). The complement rule is often easier to use than computing a probability directly. For example, P(at least one adverse event) = 1 - P(no adverse events).
IV. Addition Rule
The general addition rule states that P(A or B) = P(A) + P(B) - P(A and B). The intersection must be subtracted to avoid double-counting outcomes that belong to both events. In the special case where A and B are mutually exclusive (disjoint) events -- meaning they cannot occur simultaneously -- P(A and B) = 0, and the rule simplifies to P(A or B) = P(A) + P(B). For example, a patient cannot be both blood type A and blood type B, so these are mutually exclusive events.
As a clinical example, P(patient has diabetes OR hypertension) = P(diabetes) + P(hypertension) - P(both), because some patients may have both conditions.
<image>Two Venn diagrams illustrating the addition rule. Panel A: Two overlapping circles representing events A and B within a rectangle (sample space S). The overlap region is shaded and labeled "P(A and B)." The formula P(A or B) = P(A) + P(B) - P(A and B) is written below. Panel B: Two non-overlapping circles representing mutually exclusive events, with no shaded overlap. The formula simplifies to P(A or B) = P(A) + P(B).</image>
V. Multiplication Rule and Independence
The general multiplication rule states that P(A and B) = P(A) P(B|A), where P(B|A) is the conditional probability of B given A. Two events are independent if the occurrence of one does not affect the probability of the other, meaning P(B|A) = P(B), which simplifies the rule to P(A and B) = P(A) P(B). Dependent events are those where the occurrence of one changes the probability of the other, requiring the use of conditional probability.
As a clinical example, if two diagnostic tests are independent, then P(both positive) = P(test 1 positive) * P(test 2 positive). An important distinction to remember is that mutually exclusive events are NOT independent -- if A occurs, B cannot occur, so knowing A happened changes the probability of B to zero.
VI. Counting Methods
The multiplication principle states that if task 1 can be done in m ways and task 2 in n ways, both tasks can be done in m n ways. Permutations count ordered arrangements of objects, with the formula nPr = n! / (n - r)!, and are used when order matters. Combinations count unordered selections, with the formula nCr = n! / [r! (n - r)!], and are used when order does not matter. For example, choosing 3 patients from 10 for a focus group yields 10C3 = 120 possible groups. Counting methods are essential for computing probabilities in situations with large sample spaces.
<image>A visual comparison of permutations vs. combinations using a simple clinical example. Panel A (Permutations): Selecting 2 patients from {A, B, C} in order, showing all 6 arrangements (AB, AC, BA, BC, CA, CB) connected by arrows. Panel B (Combinations): Selecting 2 patients from {A, B, C} without regard to order, showing only 3 groups ({A,B}, {A,C}, {B,C}). Below each, the formula and numerical result are displayed.</image>
VII. Probability Trees
A tree diagram organizes sequential events and their probabilities. Each branch represents a possible outcome with its associated probability. The probability of any path through the tree equals the product of probabilities along that path, and the sum of probabilities of all final branches equals 1. Tree diagrams are particularly useful for multi-stage clinical decision problems, such as calculating the probability of a true positive result through sequential testing.
VIII. Clinical Applications of Probability
Probability concepts have wide-ranging clinical applications. They are used to estimate the probability of disease given symptoms, to calculate the chance of an adverse event over multiple doses, and to determine the probability that at least one test in a panel yields a falsely positive result. For instance, if 20 independent tests are run at alpha = 0.05, P(at least one false positive) = 1 - (0.95)^20 = 0.64. This surprisingly high probability motivates corrections for multiple comparisons.
Genetic probability also relies on classical probability. In Mendelian inheritance, if both parents are carriers of an autosomal recessive trait, the probability that their child has the disease is 1/4.
<image>A probability tree for a two-stage diagnostic scenario. First branch: disease present (prevalence = 0.10) vs. disease absent (0.90). Second branches: test positive or negative, with sensitivity = 0.95 and specificity = 0.90 annotated on the appropriate branches. Final outcomes are labeled: true positive, false negative, false positive, true negative. Joint probabilities are calculated at each terminal node, and the total probability of a positive test is summed at the bottom.</image>
