🫁Intro to Biostatistics Unit 2 – Probability Theory

Key Concepts and Definitions

• Probability the likelihood of an event occurring, expressed as a number between 0 and 1
  • 0 indicates an impossible event, while 1 represents a certain event
• Sample space the set of all possible outcomes of an experiment or random process
• Event a subset of the sample space, representing one or more outcomes of interest
• Random variable a function that assigns a numerical value to each outcome in a sample space
  • Can be discrete (countable values) or continuous (uncountable values)
• Independence two events are independent if the occurrence of one does not affect the probability of the other
• Mutually exclusive events cannot occur simultaneously (rolling a 1 and a 6 on a single die roll)

Probability Basics

• Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes
  • P(A) = (number of favorable outcomes) / (total number of possible outcomes)
• The sum of probabilities for all possible outcomes in a sample space equals 1
• Complement of an event (A') probability that event A does not occur, calculated as P(A') = 1 - P(A)
• Addition rule for mutually exclusive events P(A or B) = P(A) + P(B)
• Multiplication rule for independent events P(A and B) = P(A) × P(B)
• Conditional probability the probability of event A occurring given that event B has already occurred, denoted as P(A|B)
  • Calculated as P(A|B) = P(A and B) / P(B)

Types of Probability

• Classical probability based on the assumption that all outcomes are equally likely
  • Used in situations with a finite number of equally likely outcomes (fair coin, unbiased die)
• Empirical (frequentist) probability estimated based on observed data or past experiences
  • Calculated as the relative frequency of an event in a large number of trials
• Subjective probability based on personal belief or judgment, often used when limited data is available
• Axiomatic probability follows a set of axioms to ensure consistency and avoid paradoxes
  • Non-negativity P(A) ≥ 0 for all events A
  • Normalization P(S) = 1, where S is the entire sample space
  • Additivity for mutually exclusive events P(A or B) = P(A) + P(B)

Probability Distributions

• Probability distribution a function that describes the likelihood of different outcomes for a random variable
• Discrete probability distributions used for random variables with countable outcomes
  • Examples Bernoulli, binomial, Poisson, geometric distributions
• Continuous probability distributions used for random variables with uncountable outcomes
  • Examples uniform, normal (Gaussian), exponential, beta distributions
• Probability density function (PDF) describes the relative likelihood of a continuous random variable taking on a specific value
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value
• Expected value (mean) the average value of a random variable over a large number of trials
• Variance and standard deviation measures of the spread or dispersion of a probability distribution

Applications in Biostatistics

• Diagnostic testing calculating sensitivity, specificity, and predictive values using probability
  • Sensitivity P(positive test | disease), specificity P(negative test | no disease)
• Epidemiology estimating disease prevalence, incidence, and risk factors using probability methods
• Genetics calculating the probability of inheriting certain traits or genetic disorders based on Mendelian inheritance
• Clinical trials determining the probability of treatment success, adverse events, and patient outcomes
• Survival analysis estimating the probability of survival over time using methods like Kaplan-Meier curves and Cox regression
• Risk assessment quantifying the probability of developing a disease or experiencing an adverse event based on risk factors

Probability Calculations

• Bayes' theorem used to calculate the probability of an event based on prior knowledge and new evidence
  • P(A|B) = (P(B|A) × P(A)) / P(B)
• Permutations calculate the number of ways to arrange objects in a specific order
  • nPr = n! / (n - r)!, where n is the total number of objects and r is the number of objects being arranged
• Combinations calculate the number of ways to select objects without regard to order
  • nCr = n! / (r! × (n - r)!), where n is the total number of objects and r is the number of objects being selected
• Binomial probability calculates the probability of a specific number of successes in a fixed number of independent trials
  • P(X = k) = nCk × p^k × (1 - p)^(n - k), where n is the number of trials, k is the number of successes, and p is the probability of success in a single trial
• Poisson probability calculates the probability of a specific number of events occurring in a fixed interval of time or space
  • P(X = k) = (λ^k × e^(-λ)) / k!, where λ is the average number of events per interval and k is the number of events of interest

Common Mistakes and Pitfalls

• Confusing independence and mutual exclusivity events can be mutually exclusive but not independent, or independent but not mutually exclusive
• Misinterpreting conditional probability P(A|B) is not always equal to P(B|A)
• Neglecting the base rate (prior probability) when using Bayes' theorem
• Misusing the multiplication rule for non-independent events P(A and B) ≠ P(A) × P(B) if A and B are dependent
• Overestimating the likelihood of rare events based on personal experience or media coverage (availability heuristic)
• Misinterpreting p-values as the probability of the null hypothesis being true, rather than the probability of observing the data given that the null hypothesis is true

Real-World Examples

• Weather forecasting predicting the probability of rain, snow, or other weather events based on historical data and current conditions
• Insurance calculating premiums based on the probability of claims, considering factors like age, health status, and risk behaviors
• Quality control estimating the probability of defective products in a manufacturing process to ensure compliance with standards
• Sports betting determining the odds of different outcomes in a game or tournament based on team statistics and performance
• Medical decision-making using probability to weigh the risks and benefits of different diagnostic tests or treatment options
• Finance assessing the probability of investment returns, loan defaults, or market fluctuations to inform financial strategies