Introduction to Probability & Statistics Concepts of Probability

1. South Dakota School of Mines & TechnologyIntroduction to Probability & StatisticsIndustrial Engineering

2. Introduction to Probability & StatisticsConcepts of Probability

3. Probability Concepts S = Sample Space : the set of all possible unique outcomes of a repeatable experiment. Ex: flip of a coin S = {H,T} No. dots on top face of a die S = {1, 2, 3, 4, 5, 6} Body Temperature of a live human S = [88,108]

4. Probability Concepts Event: a subset of outcomes from a sample space. Simple Event: one outcome; e.g. get a 3 on one throw of a die A = {3} Composite Event: get 3 or more on throw of a die A = {3, 4, 5, 6}

5. Rules of Events Union: event consisting of all outcomes present in one or more of events making up union. Ex: A = {1, 2} B = {2, 4, 6} A  B = {1, 2, 4, 6}

6. Rules of Events Intersection: event consisting of all outcomes present in each contributing event. Ex: A = {1, 2} B = {2, 4, 6} A  B = {2}

7. Rules of Events Complement: consists of the outcomes in the sample space which are not in stipulated event Ex: A = {1, 2} S = {1, 2, 3, 4, 5, 6} A = {3, 4, 5, 6}

8. Rules of Events Mutually Exclusive: two events are mutually exclusive if their intersection is null Ex: A = {1, 2, 3} B = {4, 5, 6} A  B = { } = 

9. Probability Defined • Equally Likely Events If m out of the n equally likely outcomes in an experiment pertain to event A, then p(A) = m/n

10. Probability Defined • Equally Likely Events If m out of the n equally likely outcomes in an experiment pertain to event A, then p(A) = m/n Ex: Die example has 6 equally likely outcomes: p(2) = 1/6 p(even) = 3/6

11. Probability Defined • Suppose we have a workforce which is comprised of 6 technical people and 4 in administrative support.

12. Probability Defined • Suppose we have a workforce which is comprised of 6 technical people and 4 in administrative support. P(technical) = 6/10 P(admin) = 4/10

13. Rules of Probability Let A = an event defined on the event space S 1. 0 < P(A) < 1 2. P(S) = 1 3. P( ) = 0 4. P(A) + P( A ) = 1

14. Addition Rule P(A B) = P(A) + P(B) - P(A  B) A B

15. Addition Rule P(A B) = P(A) + P(B) - P(A  B) A B

16. Example • Suppose we have technical and administrative support people some of whom are male and some of whom are female.

17. Example (cont) • If we select a worker at random, compute the following probabilities: P(technical) = 18/30

18. Example (cont) • If we select a worker at random, compute the following probabilities: P(female) = 14/30

19. Example (cont) • If we select a worker at random, compute the following probabilities: P(technical or female) = 22/30

20. Example (cont) • If we select a worker at random, compute the following probabilities: P(technical and female) = 10/30

21. È = + - Ç P ( T F ) P ( T ) P ( F ) P ( T F ) Example (cont) • Alternatively we can find the probability of randomly selecting a technical person or a female by use of the addition rule. = 18/30 + 14/30 - 10/30 = 22/30

22. Operational Rules Mutually Exclusive Events: P(A B) = P(A) + P(B) A B

23. A Conditional Probability Now suppose we know that event A has occurred. What is the probability of B given A? A  B P(B|A) = P(A  B)/P(A)

24. Example • Returning to our workers, suppose we know we have a technical person.

25. Example • Returning to our workers, suppose we know we have a technical person. Then, P(Female | Technical) = 10/18

26. Example • Alternatively, P(F | T) = P(F T) / P(T) = (10/30) / (18/30) = 10/18 Ç

27. Independent Events • Two events are independent if P(A|B) = P(A) or P(B|A) = P(B) In words, the probability of A is in no way affected by the outcome of B or vice versa.

28. Example • Suppose we flip a fair coin. The possible outcomes are H T The probability of getting a head is then P(H) = 1/2

29. Example • If the first coin is a head, what is the probability of getting a head on the second toss? H,H H,T T,H T,T P(H2|H1) = 1/2

30. Example • Suppose we flip a fair coin twice. The possible outcomes are: H,H H,T T,H T,T P(2 heads) = P(H,H) = 1/4

31. Example • Alternatively P(2 heads) = P(H1  H2) = P(H1)P(H2|H1) = P(H1)P(H2) = 1/2 x 1/2 = 1/4

32. Tech Admin Male 8 8 Female 10 4 Example • Suppose we have a workforce consisting of male technical people, female technical people, male administrative support, and female administrative support. Suppose the make up is as follows

33. Tech Admin Male 8 8 Female 10 4 Example Let M = male, F = female, T = technical, and A = administrative. Compute the following: P(M  T) = ? P(T|F) = ? P(M|T) = ?

35. South Dakota School of Mines & TechnologyIntroduction to Probability & StatisticsIndustrial Engineering

36. Introduction to Probability & StatisticsCounting

37. Fundamental Rule • If an action can be performed in m ways and another action can be performed in n ways, then both actions can be performed in m•n ways.

38. Fundamental Rule • Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?

39. Fundamental Rule • Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? 1 2 3 4 5

40. Fundamental Rule • Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? 2 3 4 5 1 2 3 4 5

41. 3 4 5 2 3 4 5 1 2 3 4 5 Fundamental Rule • Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?

42. 3 4 5 2 3 4 5 1 2 3 4 5 Fundamental Rule • Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there? LN = 5•4•3 = 60

43. Combinations • Suppose we flip a coin 3 times, how many ways are there to get 2 heads?

44. Combinations • Suppose we flip a coin 3 times, how many ways are there to get 2 heads? Soln: List all possibilities: H,H,H H,T,T H,H,T H,T,H H,T,H T,H,H T,H,H T,T,T

45. Combinations Of 8 possible outcomes, 3 meet criteria H,H,H H,T,T H,H,T H,T,H H,T,H T,H,H T,H,H T,T,T

46. Combinations If we don’t care in which order these 3 occur H,H,T H,T,H T,H,H Then we can count by combination.

47. Combinations • Combinations nCk = the number of ways to count k items out n total items order not important. n = total number of items k = number of items pertaining to event A

48. Example • How many ways can we select a 4 person committee from 10 students available?

49. Example • How many ways can we select a 4 person committee from 10 students available? No. Possible Committees =

50. Example • We have 20 students, 8 of whom are female and 12 of whom are male. How many committees of 5 students can be formed if we require 2 female and 3 male?