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Chapter 4 Introduction to Probability Dr. Ayham Jaaron. Dealing with Uncertainly. In the previous chapters we discussed the analysis of decisions where uncertainty was not considered to be a major factor. As we have noticed that decision making involves some extent of uncertainty.
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Chapter 4 Introduction to Probability Dr. Ayham Jaaron
Dealing with Uncertainly • In the previous chapters we discussed the analysis of decisions where uncertainty was not considered to be a major factor. • As we have noticed that decision making involves some extent of uncertainty. • Buying an expensive machine might involves issues concerning reliability, failure, life span, resale value...etc. • An individual who needs to decide whether to have house insurance or not may be uncertain as to whether the house will be damaged, burgled, or flooded.
Uncertainty...con’t • This uncertainty can be measured and expressed in different forms: Simplest: words such as unlikely, impossible, probable, expected....etc. • Numbers offers a better measure for uncertainty. • For this reason we will use the concept of probability in our decision models. • Probability runs on a scale from 0 to 1.
Outcomes and events If a company is thinking of launching two new products, A and B. Then possible things that can happen are: • Both products fail. • Product A succeeds but B fails. • Product A fails but B succeeds. • Both products succeed. All possible things that can happen are called outcomes, while an event consists of one or more outcomes. Example: the event that just one product succeeds consists of two outcomes.
Approaches to probability • The classical approach: In order to apply the classical approach to a problem we have to assume that each outcome is equally likely to occur. E.g. If you have 200 components packed in a box and you know that 80 are damaged beyond repair and 30 are slightly damaged. What are the chances that you Select a component which is damaged beyond repair?
Approaches to Probability • The relative frequency approach • In the relative frequency approach the probability of an event occurring is regarded as the proportion of times that the event occurs in the long run if stable conditions apply. • For example: A quality control inspector might test 250 light bulbs and find that only eight are defective. This would suggest that the probability of a bulb being defective is 8/250. • The reliability of the inspector’s probability estimate would improve as he gathered more data: an estimate based on a sample of 10 bulbs would be less reliable than one based on the sample of 250.
Approaches to Probability • The subjective approach: • A subjective probability is an expression of an individual’s degree of belief that a particular event will occur. • Example: a sales manager may say: ‘I estimate that there is a 0.75 probability that the sales of our new product will exceed $2 million next year. • These judgments are usually accused of poor quality as it depends solely on individual opinions.
Mutually exclusive events Two events are mutually exclusive (or disjoint) if the occurrence of one of the events precludes the simultaneous occurrence of the other
The addition rule • If A and B are mutually exclusive events: p(A or B) = p(A) + p(b) For example, we may need to calculate the probability that a new product development will take either 3 or 4 years • If A and B are not mutually exclusive: p(A or B) = p(A) + p(b) – p(A and B)
Example on addition rule • For example, suppose that a manager estimates the following probabilities for the time that a new product will take to launch: • Time to launch product Probability 1 year 0.1 2 years 0.3 3 years 0.4 4 years 0.2 Determine the probability that the launch will take either 1 or 2 years? Clearly, both events are mutually exclusive so: p(launch takes 1 or 2 years) = p(takes 1 year) + p(takes 2 years) Determine the probability that the launch will take at least 2 years? p(launch takes 2 or 3 or 4 years)
Complementary events If A is an event then the event ‘A does not occur’ is said to be the complement of A.
Marginal probabilities p(worker contracts cancer) = 268/100 = 0.268 • This probability is called an unconditional or marginal because it is not conditional on whether or not the worker was exposed to the chemical.
Conditional probabilities p(worker contracts cancer | exposed to chemical) = 220/355 = 0.620
Independent events • Two events, A and B, are said to be independent if the probability of event A occurring is unaffected by the occurrence or non-occurrence of event B. If two events, A and B, are independent: p(A | B) = p(A)
The multiplication rule If A and B are independent events: p(A and B) = p(A) p(B) If A and B are not independent: p(A and B) = p(A) p(B | A)
Probability Trees • A large multinational company is concerned that some of its assets may be nationalized after that country’s next election. It is estimated that there is a 0.6 probability that the Socialist Party will win the next election and a 0.4 probability that the Conservative Party will win. If the Socialist Party wins then it is estimated that there is a 0.8 probability that the assets will be nationalized, while the probability of the Conservatives nationalizing the assets is thought to be only 0.3. The company wants to use probability tree to estimate the probability that their assets will be nationalized after the election.
The axioms of probability theory Axiom 1: Positiveness The probability of an event occurring must be non-negative. Axiom 2: Certainty The probability of an event which is certain to occur is 1. Thus axioms 1and 2 imply that the probability of an event occurring must be at least zero and no greater than 1. Axiom 3: Unions If events A and B are mutually exclusive then: p(A or B) = p(A) + p(B)
