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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Chapter 11: Probability

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Objectives Define probability Work with mutually exclusive events Work with independent events Work with dependent events Calculate expected monetary values Recognise and calculate binomial probabilities

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Defining Probability Most commonly used definition: P(Event) = Number of ways event can occur Number of outcomes BUT The problem is that not all outcomes are necessarily equally likely or equally probable (Think of a biased dice) For example, a coin has 2 sides, so probability of a Head is 1/2

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage More Definitions Frequency definition: P(Event) = Number of times event occurs Number of trials Subjective definition: Ask a series of “experts” to estimate the probability and use feedback to refine the figure Axiomatic approach: Set up axioms and derive probability results from these

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Mutually Exclusive Events Rule 1: Where A and B are mutually exclusive, then: P(A or B) = P(A) + P(B) If two (or more) events cannot happen at the same time, they are called mutually exclusive. This could be different scores on a die, since only one face can show at a time. In this case we can just add the probabilities together: P(1) + P(2) = 1/6 + 1/6 = 1/3 = P(1 or 2)

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Non-Mutually Exclusive Events Some items or events have more than one characteristic If you think of a pack of cards, each card has a value and a suit. Rule 2: Where A and B are not mutually exclusive, then: P(A or B) = P(A) + P(B) – P(A and B) Sample space:

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Independent Events Rule 3 Where A and B are independent, then: P(A & B) = P(A) x P(B) If the outcome of one event does not affect the outcome of some other event, then the 2 events are independent For example, if two coins are tossed, the result on the first coin has no effect on the result on the second coin So we can multiply the probabilities P(H on 1st) x P(H on 2nd) = ½ x ½=¼ = P(H on both coins)

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Dependent Events Where the outcome of one event affects the outcome of a second event, the second event is called dependent For example, if we have a small group of people consisting of 3 men and 4 women and want to find probabilities relating to selecting 2 people, then the probability relating to the second person will depend on who was selected first. First Person P(Man) = 3/7 P(Woman) = 4/7 2nd person if 1 st Male P(Man) = 2/6 P(Woman) = 4/7 2nd person if 1 st Female P(Man) = 3/6 P(Woman) = 3/7

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Expected Values Probability is a useful concept, but many people find it a little difficult, unless they are doing a course such as this. They will want to know what the effect or outcome is, and we can help by talking about what is expected to happen This expectation is a sort of “average in the long run” And we find it by multiplying the effect by the probability So, if we have a lottery ticket which might win £1,000 with a probability of 0.001, but win nothing otherwise, then we have the expected value of the ticket as: £1,000 x 0.001 + £0 x 0.999 = £1 Don’t think this is what will happen – it can’t! It is the average value of the ticket, so don’t pay £2 for it!

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Decision Trees This is a way of illustrating the consequences of decisions and the outcomes of events for which we can assign probabilities As a visual medium, it can help communicate ideas to other people It can also help you to think through a problem, making sure that you have considered every option. When using decision trees we are usually dealing with monetary outcomes, so rather than expected values, we usually talk about Expected Monetary Values

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Decision Trees (2) Think about the following example: A company is deciding whether or not to expand. If they expand, at a cost of £50,000 and the market also expands, their expected increase in profit is £100,000 per year. Expansion without an increase in the market might give increased profits of £20,000 per year. If they do not expand and the market grows, they will still increase their profits by £20,000 per year, but if the market does not grow, then there is no foreseeable increase in profits. The probability the market grows as predicted is 0.25

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Decision Trees (3) For the expand company option, we have: Market expands p=0.25 Expected Profit Market doesn’t expand p=0.75 £100,000*.25 =£25,000 £20,000*.75 =£15,000 The overall expected profit increase is £25,000 + £15,000 = £40,000

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Decision Trees (4) For the do not expand company option, we have: Market expands p=0.25 Expected Profit Market doesn’t expand p=0.75 £20,000*.25 =£5,000 £0*.75 =£0 The overall expected profit increase is £5,000 + £0 = £5,000

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Decision Trees (5) Putting the two diagrams together, we get: Expand Cost=£50,000 Don’t expand Cost = £0 Market grows Expected Profit £25,000 No growth £15,000 Market grows £5,000 £0 No growth

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Decision Trees (6) Looking at the overall outcomes If the company expands the cost is £50,000 and the expected increase in profit is £40,000 a net expected cost of £10,000 If the company does not expand, the cost is £0 and the expected increase in profit is £5,000 a net expected profit of £5,000 You would recommend the company not to expand

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Binomial Probabilities A binomial model will apply where (a) the chance of success (however defined) is constant from event to event (b) there are two outcomes (or at least, the outcomes can be classified into just two) – success and failure Binomial models have proved useful in solving many problems and are fairly easy to use.

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Binomial Results If there are two trials, where the probability of success is 0.4, then there are 4 possible results: ResultCalculationProbability Two successes.0.4 x 0.40.16 Success on the first trial and failure on the second. 0.4 x 0.60.24 Failure on the first trial and success on the second 0.6 x 0.40.24 Two failures0.6 x 0.60.36 Total1.00

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Combining Results Looking at the last table, you can see that the middle two results are the same, except for the order in which the events occur Since order doesn’t matter, we can combine these to get: ResultCalculationProbability Two successes.0.4 x 0.40.16 One success2 x 0.4 x 0.60.48 Zero successes0.6 x 0.60.36 Total1.00

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Binomial Formula We can derive a general formula for binomial situations where n = number of trials and p = probability of success and r = number of successes The number of different ways of getting r successes is given by:

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Binomial Example If we pick 4 items at random from a production process where we know that the probability of a faulty item is constant at 0.1 What is the probability of getting 2 faulty items in the sample? In this case we can identify that p = 0.1, n = 5 and r = 2 Putting these into our formula gives: The probability will be:

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Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Conclusions Probability provides a very useful means of looking at problems. It highlights the fact that little is absolutely certain and that alternative outcomes have some chance of happening. Learning to think using probability will help you to consider areas such as contingency planning and risk analysis.

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