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Delve into the fundamental terms of probability with this comprehensive guidebook, covering conditional probability, Bayes’ theorem, and real-world applications. Learn about random circumstances, sample spaces, basic probability rules, set theory terms, handling independent events, and more! Enhance your understanding of probability through clear explanations, illustrative examples, and practical tools for calculating probabilities in various scenarios.
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When you encounter a challenge, just do your best, then accept the results and finally let it go.
Probability Basic terms Conditional probability Bayes’ theorem Application
Random Circumstance • A random circumstance is one in which the outcome is unpredictable. The outcome is unknown until we observe it.
Basic Terms • Sample space S: the collection of all possible outcomes of a random circumstance • An event is a collection of one or more outcomes in the sample space. The event A is said to “occur” if the outcome falls in the set A. So each time an outcome is observed, a given event A either occurs or does not occur.
Definition of Probability • P(A) = the probability of the event A = the long-run relative frequency of occurrence of the event A = the proportion or fraction of times that the event A occurs in a large number of independent identical random circumstances. • P(A) can also represent the likelihood of occurrence of A based on a subjective assessment. (eg. What is the probability that a flipped coin shows heads up?)
Basic Probability Rules • Between 0 and 1 • The sum of the probabilities over all possible outcomes is 1 • The sum of the probabilities of simple events in the event
Equally Likely Probability Model • If the sample space S is finite in number and the outcomes have the same likelihood of occurrence, then each outcome has probability equal to 1 divided by the number of possible outcomes and so
Basic Set Theory Terms • A or B = AUB = the union of A and B = the set of all outcomes that are in A or in B or in both A and B. • A and B = A∩B = the intersection of A and B = the set of all outcomes that are in both A and B. • Not A = Ac = the complement of A = the set of all outcomes that fall outside of A. • A and B are disjoint or mutually exclusive if is the empty set; that is, A and B have no outcomes in common. Therefore, A and B are mutually exclusive if and only if they cannot occur together. * Show Venn diagram
More Probability Rules • Addition rule: P(A U B) = P(A) + P(B) - P(A and B) • P(A1 U A2 U…U Ak)=P(A1)+P(A2)+…+P(Ak) if A1, A2, …, Ak are mutually exclusive events (no two events can occur together). • Complement rule: P(Ac) = 1-P(A)
Conditional Probability • When we calculate P(A), the probability of an event A, we assume that we know nothing about the outcome of the random experiment except that it will be a member of the sample space. But when we calculate P(A | B), the conditional probability of an event A given the event B, we assume that the outcome will be a member of the event B. That is, we are given some partial information about the outcome (it will fall in B) and we have to calculate the probability that the outcome will also fall in A. Sometimes, P(A | B) can be calculated using our intuition. In other situations, we may have to use the formula:
Independent Events • Knowing that B will occur may or may not affect the probability of A. If the occurrence of B does not change the probability of A, then A and B are said to be independent; otherwise, they are dependent. • So A and B are independent if and only if P(A|B)=P(A) Note that P(A|B)=P(A) is equivalent to P(B|A)=P(B).
More Probability Rules • Multiplication rule: P(A ∩ B) = P(A) x P(B|A) = P(B) x P(A|B) • P(A1 ∩ A2 ∩…∩ Ak)=P(A1)xP(A2)x…xP(Ak) if A1, A2, …, Ak are independent events
Tools for Finding Probabilities • When conditional or joint probabilities are known for two events Two-way tables • For a sequence of events, when conditional probabilities for events based on previous events are known Tree diagrams
Total Probability Law • Assume the sample space S is “partitioned” by the events A1, A2, …, Ak meaning that • 1) and • 2) if i ≠ j (the events are pairwise disjoint). • This says that the outcome of the random circumstance will be in one and only one of the partitioning events. • Total Probability Law asserts that for any event B,
Bayes’ Theorem Think of the events A1, A2,…, Ak as representing all possible conditions that can produce the observable “effect” B. In this context, the probabilities P(Ai)’s are called prior probabilities. Now suppose that the effect B is observed to occur. Bayes’ theorem gives a way to calculate the probability that B was produced or caused by the particular condition Ai than by any of the other conditions. The conditional probability P(Ai|B) is called the posterior probability of Ai .
Diagnostic Tests • A diagnostic testing or screening is the application of a test to individuals who have not yet exhibited any clinical symptoms in order to classify them with respect to their probability of having a particular disease. • “sensitivity” is the true positive rate • “specificity” is the true negative rate • “prevalence” is the proportion of subjects with the disease in a population
Relative Risk • The relative risk, RR, is the chance that a subject receiving some exposure will develop disease relative to the chance that an unexposed subject will develop the same disease.
Odds and Odds Ratio • If an event (disease) takes place with chance p, the odds of the event are p/(1-p), the ratio of the chance the event occurs to the chance it does not occur. • The odds ratio (OR) of a disease for exposed subjects to unexposed subjects is:
Example: Smoke vs. Lung Cancer • In US, the chance that a man over the age of 35 dies of lung cancer is .002679 for current smokers and .000154 for nonsmokers • RR=? • OR=?
