1.  Review Homework Chapter 6: 1, 2, 3, 4, 13 Chapter 7 - 2, 5, 11  Probability  Control charts for attributes  Week 13 Assignment Read Chapter 10: “Reliability” Homework Chapter 8: 5, 9,10, 20, 26, 33, 34 Chapter 9: 9, 23 Week 12 Agenda

2. Probability Probability Chapter Eight

3. Probability Probability theorems  Probability is expressed as a number between 0 and 1  Sum of the probabilities of the events of a situation equals 1  If P(A) is the probability that an event will occur, then the probability the event will not occur is 1.0 - P(A)

4. Probability Probability theorems  For mutually exclusive events, the probability that either event A or event B will occur is the the sum of their respective probabilities.  When events A and B are not mutually exclusive events, the probability that either event A or event B will occur is P(A or B or both) = P(A) + P(B) - P(both)

5. Probability Probability theorems  If A and B are dependent events, the probability that both A and B will occur is P(A and B) = P(A) x P(B|A)  If A and B are independent events, then the probability that both A and B will occur is P(A and B) = P(A) x P(B)

6. Probability Permutations and combinations  A permutation is the number of arrangements that n objects can have when r of them are used.  When the order in which the items are used is not important, the number of possibilities can be calculated by using the formula for a combination.

7. Probability Discrete probability distributions  Hypergeometric - random samples from small lot sizes. Population must be finite samples must be taken randomly without replacement  Binomial - categorizes “success” and “failure” trials  Poisson - quantifies the count of discrete events.

8. Probability Continuous probability distributions  Normal  Uniform  Exponential  Chi Square  F  student t

9. Probability Fundamental concepts  Probability = occurrences/trials  0 < P < 1  The sum of the simple probabilities for all possible outcomes must equal 1  Complementary rule - P(A) + P(A’) = 1

10. Probability Addition rule  P(A + B) = P(A) + P(B) - P(A and B) If mutually exclusive; just P(A) + P(B) P(A)P(B) P(AandB)

11. Probability Addition rule example  P(A + B) = P(A) + P(B) - P(A and B)  Roll one die Probability of even and divisible by 1.5? Sample space {1,2,3,4,5,6} Event A - Even {2,4,6} Event B - Divisible by 1.5 {3,6} Event A and B ?  Solution?

12. Probability Conditional probability rule  P(A|B) = P(A and B) / P(B)  A die is thrown and the result is known to be an even number. What is the probability that this number is divisible by 1.5? P(/1.5|Even)=P(/1.5 and even)/P(even) 1/6 / 3/6 = 1/3

13. Probability Compound or joint probability  The probability of the simultaneous occurrence of two or more events is called the compound probability or, synonymously, the joint probability.  Mutually exclusive events cannot be independent unless one of them is zero.

14. Probability Multiplication for independent events  P(A and B) = P(A) x P(B) P(ace and heart) = P(ace) x P(heart) 1/13 x 1/4 = 1/52

15. Probability Computing conditional probabilities  P(A|B) = P(A and B)/P(B)  P(Own and Less than 2 years)?

16. Probability P(A)P(B) P(AandB) Computing conditional probabilities  P(A|B) = P(A and B)/P(B)

17. Probability

18. Conditional probability Satisfied Not Satisfied Totals New 300100400 Used 450150600 Total 7502501000 S=satisfied N= bought new car P(N|S) = ?

19. Probability Just for fun  60 business students from a large university are surveyed with the following results: 19 read Business Week 18 read WSJ 50 read Fortune 13 read BW and WSJ 11 read WSJ and Fortune 13 read BW and Fortune 9 read all three  How many read none?  How many read only Fortune?  How many read BW, the WSJ, but not Fortune?  Hint: Try a Venn diagram.

20. Probability Probability Distributions

21. Probability Learning objectives  Know the difference between discrete and continuous random variables.  Provide examples of discrete and continuous probability distributions.  Calculate expected values and variances.  Use the normal distribution table.

22. Probability Random variables  A random variable is a numerical quantity whose value is determined by chance. “A random variable assigns a number to every possible outcome or event in an experiment”. For non-numerical outcomes such as a coin flip you must assign a random variable that associates each outcome with a unique real number.

23. Probability Random variable types  Discrete random variable - assumes a limited set of values; non-continuous, generally countable number of Mark McGwire homeruns in a season number of auto parts passing assembly-line inspection GRE exam scores

24. Probability Random variable types  Continuous random variable - random variable with an infinite set of values. Can occur anywhere on a continuous number scale 0.000 1.000 Baseball player’s batting average

25. Probability Random variables and probability distributions  The relationship between a random variable’s values and their probabilities is summarized by its probability distribution.

26. Probability Probability distribution  Whether continuous or discrete, the probability distribution provides a probability for each possible value of a random variable, and follows these rules: The events are mutually exclusive The individual probability values are between 0 and 1. The total value of the probability values sum to 1

27. Probability Probability distribution for rates of return  Possible rate of return 10% 11% 12% 13% 14% 15% 16% 17%  Probability.05.15.20.35.10.03.02  Total = 1.0

28. Probability Describing distributions  Measures of central tendency expected value (weighted average)  Measures of variability variance standard deviation

29. Probability Expected value of a discrete random variable  For discrete random variables, the expected value is the sum of all the possible outcomes times the probability that they occur. E(X) =  {x i * P(x i )}

30. Probability Example: A fair die  Roll 1 die: x P(x) x*P(x) E(x)=? 1 1/6 1/6 2 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 6/6 Can you sketch the distribution?

31. Probability Fair die illustrates a discrete “uniform distribution”  The random variable, x, has n possible outcomes and each outcome is equally likely. Thus, x is distributed uniform.

32. Probability x P(x) 1/6 1 2 3 4 5 6 Probability distribution

33. Probability Example: An unfair die  Roll 1 die: x P(x) x*P(x) E(x)=? 1 1/12 1/12 2 2/12 4/12 3 2/12 6/12 4 2/12 8/12 5 2/12 10/12 6 3/12 18/12 Can you sketch the distribution?

34. Probability Expected value of a bet  Suppose I offer you the following wager: You roll 1 die. If the result is even, I pay you $2.00. Otherwise you pay me $1.00.  E(your winnings)=.5 ($2.00) +.5 (-1.00) = 1.00 -.50 = $0.50

35. Probability Expected Value of a Bet  Suppose I offer you the following wager: You roll 1 die. If the result is 5 or 6 I pay you $3.00. Otherwise you pay me $2.00.  What is your expected value?

36. Probability Variance of a discrete random variable The variance of a random variable is a measure of dispersion calculated by squaring the differences between the expected value and each random variable and multiplying by its associated probability.  {(x i -E(x)) 2 * P(x i )}

37. Probability  Roll 1 die: [x- E(X)] 2 P(x) *P(x) 1 - 21/6 6.25 1/6 1.04 2 - 21/6 2.25 1/6.375 3 - 21/6.25 1/6.04 4 - 21/6.25 1/6.04 5 - 21/6 2.25 1/6.375 6 - 21/6 6.25 1/6 1.04 2.91 Example: A fair die

38. Probability Probability distributions for continuous random variables  A continuous mathematical function describes the probability distribution.  It’s called the probability density function and designated ƒ(x)  Some well know continuous probability density functions: Normal Beta Exponential Student t

39. Probability Continuous probability density function - Uniform If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by ab 1 b-a

40. Probability Uniform ab 1 b-a What is the probability of x? x

41. Probability Uniform ab 1 b-a Area under the rectangle = base*height = (b-a)* 1 = 1 b-a

42. Probability Uniform ab 1 b-a c P(c<x<b) = Area of brown rectangle 1 * (b-c) Ht x Width) = b-a

43. Probability Uniform 15 1 5-1 2 P(2<x<5) = Brown rectangle 1 * (5-2) =(1/4) *3 =  3/4 = 5-1 = 1/4

44. Probability Uniform distribution If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by And, the mean and variance are (a+b) ( b-a ) 2 E(x) = ------- Var(x)=--------- 2 12

45. Probability Uniform 38 Mean? Variance?

46. Probability And, the mean and variance are (a+b) ( b-a ) 2 25 E(x) = ------ = 5.5 V(x)=--------- = ----- = 2.08 2 12 12 So, if a = 3 and b = 8 Calculate uniform mean, variance

47. Probability Continuous pdf - Normal If x is a normally distributed variable, then is the pdf for x. The expected value is  and the variance is  2.

48. Probability One standard deviation 68.3% 

49. Probability Two standard deviations 95.5% 22 22

50. Probability Three standard deviations 99.73% 33 33

51. Probability Continuous PDF - Standard Normal If z is distributed standard normal, then  and 

52. Paper Review Probability -http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.509 6http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.509 6 -http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.2335http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.2335 -http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.102.844http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.102.844 -http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.3884http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.3884