1. Welcome to MM305 Unit 3 Seminar Prof Greg Probability Concepts and Applications

2. The Basics of Probability Events Outcomes Probability Experiment Sample Space

3. Probability Basics Experiment: Rolling a single die Sample Space: All possible outcomes from experiment S = {1, 2, 3, 4, 5, 6} Event: a collection of one or more outcomes (denoted by capital letter) Event A = {3} Event B = {even number} Probability = (number of favorable outcomes) / (total number of outcomes) P(A) = 1/6 P(B) = 3/6 = ½

4. More Probability Basics Probability will always be between 0 and 1. It will never be negative or greater than 1. Complement of an event: All outcomes that are not included in the Event of interest. If A = {3} then the “not A” or A ’ = {1, 2, 4, 5, 6}. A ’ is everything but 3 The sum of the simple probabilities for all possible outcomes of an activity must equal 1

5. The Basics of Probability Three ways to calculate probability:  Classical Probability: Proportion of times that an event can be theoretically expected to occur. For outcomes that are equally likely to occur, Probability of Event X= ( total number of favorable outcomes for event X ) ( total number of possible outcomes ) This is the standard way to calculate probability  Relative Frequency Probability: Proportion of times that a probability is expected to occur over a large number of trials. For a very large number of trials, Probability of Event X= ( total number of trials for event X ) ( total number of trials )  Subjective Probability: Probabilities estimated by making an educated guess; based solely on belief that the event will happen

6. More Basics Concepts of Probability Independent Events Two events are said to be independent if the outcome of the second event is not affected by the outcome of the first event. They cannot influence or affect each other. Mutually Exclusive Events Two events are said to be mutually exclusive if they cannot occur at the same time. Compound Probability AND P(A and B) = P(A)*P(B) when the events are independent P(A and B) = P(A) + P(B) – P(A or B) when the events are dependent Compound Probability OR P(A or B) = P(A) + P(B) when the events are mutually exclusive P(A or B) = P(A) + P(B) – P(A and B) when the events are not mutually exclusive Conditional Probability P(B | A), event B given that event A has occurred ( P(B | A) ≠ P(A | B) ) P(B | A) = P(B) and P(A|B) = P(A) when events are independent

7. Mutually Exclusive Events Events are said to be mutually exclusive if only one of the events can occur on any one trial Tossing a coin will result in either a head or a tail Rolling a die will result in only one of six possible outcomes

8. Probability: Tying it all together 0.00% (A) 0.01-0.09% (B) ≥0.10% (C) Total 0-19 (D) 14276155 20-39 (E) 4784196 40-49 (F) 29877114 Over 60 (G) 4773589 Total26530159454 Blood Alcohol Level of Victim Age

9. Venn Diagrams P ( A ) P ( B ) Events that are mutually exclusive P ( A or B ) = P ( A ) + P ( B ) Events that are not mutually exclusive P ( A or B ) = P ( A ) + P ( B ) – P ( A and B ) P ( A ) P ( B ) P ( A and B )

10. Random Variables Discrete random variables Discrete random variables can assume only a finite or limited set of values Continuous random variables Continuous random variables can assume any one of an infinite set of values Always define what your random variable represents! Let X = number of people, companies, computers, hours, etc. A random variable assigns a real number to every possible outcome or event in an experiment

11. Numerical Descriptors of a Discrete Probability Distribution General Formulas for mean and variance: Mean (Expected Value) µ = Σ (x*P(x) ) Variance σ 2 = Σ ( (x- µ) 2 * P(x) ) Standard Deviation = σ = √σ 2 for all possible values of x

12. QM for Windows : Select Statistics

13. QM for Windows : Select Data Analysis

14. QM for Windows : Select # Values, Data Type

15. QM for Windows : Enter Values; Press Solve

16. QM for Windows : Table with Mean, Variance

17. QM for Windows : Select Window then Graph

18. Excel QM : Select Probability Distribution

19. Excel QM : Select # Values, Data Type

20. Excel QM : Enter Values => Mean, Variance

21. Binomial Distribution 1: The number of trials n is fixed. 2: Each trial is independent. 3: Each trial represents one of two outcomes ("success" or "failure"). 4: The probability of "success" p is the same for each outcome. If these conditions are met, then X has a binomial distribution with parameters n and p, denoted X~B(n, p).

22. The Binomial Distribution  Each trial has only two possible outcomes  The probability stays the same from one trial to the next  The trials are statistically independent  The number of trials is a positive integer

23. Expected Value (Mean) and Variance of The Binomial Distribution Mean (Expected Value) µ = E(x) =n*p Variance σ 2 = n* p *(1- p) Standard Deviation = √σ 2 = √n* p *(1- p) Where n = number of trials x = number of successes p = probability of success (1- p) = probability of failure

24. Binomial Distribution Suppose 12% of telemarketers make a sale on a cold call, what is the probability if 10 telemarketers make a cold call that 3 of them will make a sale? Identify what we know: n= 10 x=3 p=0.12 q=1-0.12=0.88

25. Excel Function: BINOMDIST P(X=3) = BINOMDIST(3,10,0.12,FALSE) = 0.0847 P(X<=3) = BINOMDIST(3,10,0.12,TRUE) = 0.9761 P(X>3) = 1 - P(X<=3) = 1 - 0.9761 = 0.0239 E(X)= n*p= 10*0.12=1.2 Variance σ 2 = 10* 0.12 *(0.88) =1.056 Std Deviation = √σ 2 = √1.056 = 1.0276

26. Normal Probability Distribution It is a continuous probability distribution Two values determine its shape μ = mu = mean of distribution σ = sigma = standard deviation of the distribution

27. Normal Probability Distribution Remember the Empirical Rule!!!

28. Standard Normal Distribution µ = 0 σ =1 z score – tells us how standard deviations away from the mean a value is: z = (x - µ)/ σ We convert x values to z scores using the above formula or Excel! {Standardize} 190 290 390 490 590 690 790

29. Finding Normal Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6)

30. Finding Normal Probabilities Solution to previous example…. X is normal with mean 8.0 and standard deviation 5.0, so X~N(8,5) Find P(X < 8.6) = NORMDIST(8.6,8,5,TRUE) = 0.5478 Z is std normal with mean 0 and standard deviation 1.0, so Z~N(0,1) Find P(Z < 0.12) = NORMSDIST(0.12) = 0.5478 If you want to find the value of X and Z using probabilities and you know the mean and standard deviation: Using Excel, For X value, =NORMINV(0.5478,8,5) = 8.6 For Z value, =NORMSINV(0.5478) = 0.12

31. Using Technology Excel Functions BINOMDIST NORMDIST NORMSDIST STANDARDIZE NORMINV

32. Questions?