Lecture 1b: Probability rules
Lecture One: Basic Rules of Probability - Part Two
Introduction
Overview of foundational concepts in probability.
Importance for forensic calculations, particularly in match probabilities.
Material may be a review for some students, but essential for understanding subsequent topics.
Key Terms and Definitions
Propositions
Definition: A proposition is a statement that may be true or false, affirmable or deniable.
Typically formulated in pairs, representing competing ideas.
Example: Prosecution's claim of matching DNA vs. Defense's claim of non-matching DNA.
In legal context, propositions can be about matching evidence (e.g., "Mr. Smith is a source of the DNA").
Conditional Probabilities
Definition: Probability represents uncertainty about an event's occurrence.
Expressed as a likelihood of something being true (probability on a scale of 0 to 1).
Conditional probabilities consider available information in the probability assignment.
Example: Assessing Mr. Smith's DNA match against presented evidence in court.
Alternative Propositions
Definition: Competing hypotheses presented in a judicial context.
Prosecution's proposition: the DNA matches Mr. Smith.
Defense's alternative proposition: the DNA does not match Mr. Smith.
Subjective Probability
Definition: A measure of belief in the likelihood of an event occurring, expressed as a number between 0 and 1.
Example: Expert opinions on print or DNA matches can be considered subjective probabilities.
Vocabulary and Notation
Events vs. Propositions
Events: Measurable outcomes (e.g., rolling a die).
Propositions: Predictions or hypotheses (e.g., predicting snow).
Complement of an Event: Probability that an event did not occur.
Notation: Complement represented as R bar for event R, meaning 1 - P(R).
Experiments and Sample Spaces
Definition of an Experiment
Context: An experiment results in one outcome that can’t be predicted with certainty.
Example Scenarios:
Tossing a coin (outcomes: head or tail).
Rolling a die (outcomes: 1 through 6).
Sample Space
Definition: The set of all possible outcomes from an experiment.
Coin toss: Heads, Tails.
Die roll: 1, 2, 3, 4, 5, 6.
Calculating Probability
Fundamental Rule: Probability of outcomes must sum to 1.
Example of Coin Toss: Probability of heads = 0.5; Probability of tails = 0.5.
Events: Specific collections of sample points (e.g., probability of rolling a one or three).
Steps for Calculating Probability:
Identify the event of interest.
Determine possible outcomes.
Calculate individual probabilities of outcomes.
Basic Rules of Probability
Complement Rule
Complement: P(not A) = 1 - P(A).
Example: If P(A) = 0.7, then P(not A) = 0.3.
Union of Events
Notation: U (union of event A and event B).
Formula: P(A U B) = P(A) + P(B) - P(A ∩ B), where P(A ∩ B) is the probability of both events occurring together.
Avoids double counting probabilities.
Mutually Exclusive Events
Definition: Events that cannot occur simultaneously (P(A ∩ B) = 0).
Example: Rolling an even number vs. an odd number on a die.
Conditional Probabilities
Definition: P(A | B) = P(A ∩ B) / P(B).
Represents the probability of event A occurring given that event B has occurred.
Example: Calculating joint probabilities and the implications of two events being dependent or independent.
Independent Events
Definition: Event A and Event B are independent if the occurrence of one does not influence the probability of the other.
Formula: P(A ∩ B) = P(A) × P(B).
Example: Probability of different genetic markers being independently inherited.
Laws of Probability
First Law of Probability
Definition: Probabilities range from 0 to 1.
Impossible event (0) vs. certain event (1).
Second Law of Probability
Mutually Exclusive Events: Sum of individual probabilities.
If A and B cannot occur together, then P(A or B) = P(A) + P(B).
Third Law of Probability
Non-Independently Occurring Events: Multiply probabilities together.
Example: Probability of observing multiple traits together that are dependent on one another.
Law of Total Probability
Concept: Total probability of A can be partitioned into subsets based on mutually exclusive events.
Formula: P(A) = Sum of P(A | B) × P(B) for all mutually exclusive events B.
Conclusion
Recap of importance of basic probability rules in forensic science.
Importance of having a solid foundational understanding for practical applications in DNA and forensic analysis, including match probabilities.