It is assumed that the probability of a success, p, is constant over the entire set of trials for binomial random variables. Can the binomial probability distribution be a model for determining the likelihood of any discrete random variable with two mutually exclusive and collectively exhaustive outcomes? Can you think of any real-world business scenarios where meeting the assumption that the probability of success might be difficult to meet? Solution Bayes\' Theorem Bayes\' Theorem is a result that allows new information to be used to update the conditional probability of an event. Using the multiplication rule, gives Bayes\' Theorem in its simplest form: P(A | B) = P(A n B)/P(B) = P(B | A).P(A)/P(B) Using the Law of Total Probability: P(A | B) = P(B | A).P(A)P(B | A).P(A) + P(B | A\').P(A\') where: P(A) = probability that event A occurs P(B) = probability that event B occurs P(A\') = probability that event A does not occur P(A | B) = probability that event A occurs given that event B has occurred already P(B | A) = probability that event B occurs given that event A has occurred already P(B | A\') = probability that event B occurs given that event A has not occurred already Bayes\' theorem deals with the role of new information in revising probability estimates. The theorem assumes that the probability of a hypothesis (the posterior probability) is a function of new evidence (the likelihood) and previous knowledge (prior probability). The theorem is named after Thomas Bayes (1702–1761), a nonconformist minister who had an interest in mathematics. The basis of the theorem is contained in as essay published in the Philosophical Transactions of the Royal Society of London in 1763. Bayes\' theorem is a logical consequence of the product rule of probability, which is the probability (P) of two events (A and B) happening— P(A,B)—is equal to the conditional probability of one event occurring given that the other has already occurred—P(A|B)—multiplied by the probability of the other event happening—P(B). The derivation of the theorem is as follows: P(A,B) = P(A|B)×P(B) = P(B|A)×P(A) Thus: P(A|B) = P(B|A)×P(A)/P(B). Bayes\' theorem has been frequently used in the areas of diagnostic testing and in the determination of genetic predisposition. For example, if one wants to know the probability that a person with a particular genetic profile (B) will develop a particular tumour type (A)—that is, P(A|B). Previous knowledge leads to the assumption that the probability that any individual will develop the specific tumour (P(A)) is 0.1 and the probability that an individual has the particular genetic profile (P(B)) is 0.2. New evidence establishes that the probability that an individual with the tumor—P(B|A)—has the genetic profile of interest is 0.5. Thus: P(A|B) = 0.1×0.5/0.2 = 0.25 The adoption of Bayes\' theorem has led to the development of Bayesian methods for data analysis. Bayesian methods have been defined as \"the explicit use of external evidence .

1. It is assumed that the probability of a success, p, is constant over the entire set of trials for
binomial random variables. Can the binomial probability distribution be a model for determining
the likelihood of any discrete random variable with two mutually exclusive and collectively
exhaustive outcomes? Can you think of any real-world business scenarios where meeting the
assumption that the probability of success might be difficult to meet?
Solution
Bayes' Theorem Bayes' Theorem is a result that allows new information to be
used to update the conditional probability of an event. Using the multiplication rule, gives
Bayes' Theorem in its simplest form: P(A | B) = P(A n B)/P(B) = P(B | A).P(A)/P(B) Using the
Law of Total Probability: P(A | B) = P(B | A).P(A)P(B | A).P(A) + P(B | A').P(A') where: P(A)
= probability that event A occurs P(B) = probability that event B occurs P(A') = probability that
event A does not occur P(A | B) = probability that event A occurs given that event B has
occurred already P(B | A) = probability that event B occurs given that event A has occurred
already P(B | A') = probability that event B occurs given that event A has not occurred already
Bayes' theorem deals with the role of new information in revising probability estimates. The
theorem assumes that the probability of a hypothesis (the posterior probability) is a function of
new evidence (the likelihood) and previous knowledge (prior probability). The theorem is named
after Thomas Bayes (1702–1761), a nonconformist minister who had an interest in mathematics.
The basis of the theorem is contained in as essay published in the Philosophical Transactions of
the Royal Society of London in 1763. Bayes' theorem is a logical consequence of the product
rule of probability, which is the probability (P) of two events (A and B) happening— P(A,B)—is
equal to the conditional probability of one event occurring given that the other has already
occurred—P(A|B)—multiplied by the probability of the other event happening—P(B). The
derivation of the theorem is as follows: P(A,B) = P(A|B)×P(B) = P(B|A)×P(A) Thus: P(A|B) =
P(B|A)×P(A)/P(B). Bayes' theorem has been frequently used in the areas of diagnostic testing
and in the determination of genetic predisposition. For example, if one wants to know the
probability that a person with a particular genetic profile (B) will develop a particular tumour
type (A)—that is, P(A|B). Previous knowledge leads to the assumption that the probability that
any individual will develop the specific tumour (P(A)) is 0.1 and the probability that an
individual has the particular genetic profile (P(B)) is 0.2. New evidence establishes that the
probability that an individual with the tumor—P(B|A)—has the genetic profile of interest is 0.5.
Thus: P(A|B) = 0.1×0.5/0.2 = 0.25 The adoption of Bayes' theorem has led to the development
of Bayesian methods for data analysis. Bayesian methods have been defined as "the explicit use
of external evidence in the design, monitoring, analysis, interpretation and reporting" of studies

2. (Spiegelhalter, 1999). The Bayesian approach to data analysis allows consideration of all
possible sources of evidence in the determination of the posterior probability of an event. It is
argued that this approach has more relevance to decision making than classical statistical
inference, as it focuses on the transformation from initial knowledge to final opinion rather than
on providing the "correct" inference.