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What is the Probability of an Event? Probability is the study of how likely it is that an event will occur. Chapter 6 Elementary Probability Probability terms include: • Trial – A single act by which an observation is noted, e.g. roll of a die. • Experiment - The process by which an observation is noted and consists of one or more trials. • Event – A subset of all the possible outcomes of an experiment E.g. Rolling a 6. • Sample space – The set of all feasible outcomes of an experiment. E.g. when rolling a die the sample space is: (1, 2, 3, 4, 5, 6). • Define the Probability of an event • Distinguish between the three Types of Probability: • Classical probability • Relative frequency probability • Subjective probability • Explain Set Operations and Venn Diagrams • Distinguish between Mutually Exclusive, Independent and Dependent Events www.learnnowbiz.com Use Financial Statistics 1 www.learnnowbiz.com Use Financial Statistics 2 What is the Probability of an Event? What is the Probability of an Event? For a set of all possible outcomes (S) with each outcome equally likely to occur, the probability of an event (E) occurring is equal to the number of ways that event can occur divided by the total number of possible outcomes. If an event has 0 probability it can never occur. If it has a probability of 1 it will always occur. The sum of the probabilities of each event in the sample space occurring will always equal 1. Probabilities are usually expressed in decimals or percentages. E.g. if an event has a probability of 0.52 then it has a 52% chance of occurring. www.learnnowbiz.com Use Financial Statistics 3 www.learnnowbiz.com Classical Probability – Straightforward mathematical approach assigns the chance that an event will occur to the number of possible results. Relative Frequency Probability – Examines previous data to assist with current decisions. Subjective Probability – It refers to decision making where perhaps less direct data exists. Final decisions are made on a subjective basis. i.e. When will the next election be held? Use Financial Statistics 4 Set Operations and Introduction to Venn Diagrams Three Types of Probability www.learnnowbiz.com Use Financial Statistics • A set is a collection of objects or elements. Elements are shown inside the parenthesis { }. • The set of all elements is called the sample space, usually denoted ‘S’. • The elements of the set are all the possible outcomes of an experiment. • Once an outcome has occurred it is called an event. • A subset of the sample space will contain some of the elements of S. A subset can also be called an event. 5 www.learnnowbiz.com Use Financial Statistics 6 Set Operations and Introduction to Venn Diagrams Venn Diagrams Sets can be displayed graphically through the use of Venn Diagrams. • A rectangle is drawn to represent the sample space and a point is assigned to represent each simple event. • Circles are drawn around the sample points to represent events. www.learnnowbiz.com Use Financial Statistics 7 www.learnnowbiz.com Set Operations - Union Use Financial Statistics 8 Set Operations - Union Union – The union of two events is the event containing all elements in the two sets. e.g. If X and Y are two events in the sample space S, the union of X and Y is the event containing all sample points in X and Y. The Union is written as ∪. www.learnnowbiz.com Use Financial Statistics 9 www.learnnowbiz.com Use Financial Statistics 10 Set Operations - Complement Mutually Exclusive Events The complement of a set contains all the elements is a sample which are not in the set itself. Events are mutually exclusive if the occurrence of any one of the events excludes the occurrence of others, e.g. flipping a head excludes the flipping of a tail. In other words the complement of an event happening is the event not happening. www.learnnowbiz.com Use Financial Statistics Mutually exclusive events have no overlapping area in the Venn diagram. 11 www.learnnowbiz.com Use Financial Statistics 12 Independent Events Dependent Events Two or more events are said to be independent if the occurrence or non occurrence of one of them in no way effects the occurrence or non-occurrence of the others. Two or more events are dependent when the probability of one event taking place is subject to another event taking place, e.g. the probability of rain and a cloudy sky. It is more likely to rain when the sky is cloudy. The events are unconnected, e.g. two consecutive coin tosses, the result of the first cannot impact the result of the second. When two events A and B are dependent, the probability of event A occurring given that B has already occurred is the probability of events A and B occurring together divided by the probability of event B. When two events are independent the probability that they will both occur is given by multiplying the probabilities that each will occur together: P(A and B) = P(A) x P(B) www.learnnowbiz.com Use Financial Statistics 13 www.learnnowbiz.com Use Financial Statistics 14