MANG1019 – Probability PDF

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Summary

Includes definitions, equations and notes on different probability rules....

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Why is probability important?  Probability in Finance is a tool for modelling financial markets their risks and returns.  In sales and digital marketing it is used for customer profiling  In logistics, organizations use probability to inform decision making processes  Public health, disaster management and many other decision problems are addressed by using probability  Organizations in cross disciplines use probability to inform employee retention  Betting market is completely dependent on probability Definitions  An event is a collection of one or more outcomes from a sample space.  The outcome of an experiment is a result that we observe.  An experiment is the process that results in an outcome.  The sample space is the collection of all possible outcomes of an experiment. Probability  Probability is the numerical measure of the likelihood that an event will occur  The probability of any event must be between 0 and 1, inclusively Different interpretations of probability  Pierre Laplace – Logical or Classical Probability  Jacob Bernoulli – Frequentist or Observed Probability  Thomas Bayes – Subjective Probability Logical or Classical Probability of an Event  The probability of an event A can be formally defined as:

P(A ) =

N u m b e r o f o u t c o m e s t h a t le a d t o e v e n t A T o t a l n u m b e r o f p o s s ib le o u t c o m e s

Frequentist Probability ‘Law of large numbers’  In many situations it not possible to determine the ‘logical’ probability of an event  Because most situations in business and management are not simple in the sense that one can logically determine probability, the use of historical data is the most common way of estimating objective probabilities. Judgemental Probability  Often it is not feasible or cost-effective to obtain survey data in order to estimate probability  What is the probability that a newly appointed CEO of a FTSE100 company will succeed?  This is such a complex event that historical data alone will lead to a very poor estimate of probability. Yet this is the question that the board of a FTSE100 firm must consider in order to appoint a CEO.  In situations like this, probability estimates are necessarily judgmental.

Probability Rules  As mentioned before, if E is an event, then 0 ≤ P(E) ≤ 1  The probability associated with any outcome must be between 0 and 1. 0 ≤ P(Oi) ≤ 1 for each outcome Oi  The sum of the probabilities over all possible outcomes must be equal to 1. P(O1) + P(O2) + … + P(On) = 1  The complement of P(A) is P(� ) and P(A) = 1- P(� )  The intersection of two events, A and B, denoted by P(A⋂�), is calculated by the product of the two events. This is denoted as the joint probability of two events, A and B. General Addition (OR) Rule  In general, the probability of an event A OR an event B is given by P(A or B) = P(A) + P(B) – P(A and B) Mutually Exclusive Events  When events A and B are Mutually Exclusive Events, i.e. they cannot happen simultaneously: P(A and B) = 0  Example: Suppose that A = ‘1st Class Degree’; B = ‘2.1 Degree’; A and B cannot happen together. 

Hence the Addition Rule Simplifies to P(A or B) = P(A) + P(B)

Expected Value of a Discrete Random Variable  The expected value of a random variable corresponds to the notion of the mean, or average, for a sample....