1. prob1 INTRODUCTION to PROBABILITY

2. prob2 BASIC CONCEPTS of PROBABILITY  Experiment  Outcome  Sample Space Discrete Continuous  Event

3. prob3 Interpretations of Probability  Mathematical  Empirical  Subjective

4. prob4 MATHEMATICAL PROBABILITY P(E) =

5. prob5 PROPERTIES  0 < P(E) < 1  P(E’) = 1 - P(E)  P(A or B) = P(A) + P(B) for two events, A and B, that do not intersect

6. prob6 Example A part is selected for testing. It could have been produced on any one of five cutting tools.  What is the probability that it was produced by the second tool?  What is the probability that it was produced by the second or third tool?  What is the probability that it was not produced by the second tool?

7. prob7 INDEPENDENT EVENTS  Events A and B are independent events if the occurrence of A does not affect the probability of the occurrence of B.  If A and B are independent P(A and B) = P(A)*P(B)

8. prob8 Example The probability that a lab specimen is contaminated is 0.05. Two samples are checked.  What is the probability that both are contaminated?  What is the probability that neither is contaminated?

9. prob9 DEPENDENT EVENTS  Events A and B are dependent events if they are not independent.  If A and B are independent P(A and B) = P(A)*P(B/A)

10. prob10 Example From a batch of 50 parts produced from a manufacturing run, two are selected at random without replacement? What is the probability that the second part is defective given that the first part is defective?

11. prob11 MUTUALLY EXCLUSIVE EVENTS Events A and B are mutually exclusive if they cannot occur concurrently. If A and B are mutually exclusive, P(A or B) = P(A) + P(B)

12. prob12 NON MUTUALLY EXCLUSIVE EVENTS If A and B are not mutually exclusive, P(A or B) = P(A) + P(B) - P(A and B)

13. prob13 Example Disks of polycarbonate plastic from a supplier are analyzed for scratch resistance and shock resistance. For a disk selected at random, what is the probability that it is high in shock or scratch resistance? Shock Resistance highlow Scratch R high 809 low 65

14. prob14 RANDOM VARIABLES  Discrete  Continuous

15. prob15 DISCRETE RANDOM VARIABLES  Maps the outcomes of an experiment to real numbers  The outcomes of the experiment are countable. Examples Equipment Failures in a One Month Period Number of Defective Castings

16. prob16 CONTINUOUS RANDOM VARIABLE Possible outcomes of the experiment are represented by a continuous interval of numbers Examples force required to break a certain tensile specimen volume of a container dimensions of a part

17. prob17 Discrete RV Example A part is selected for testing. It could have been produced on any one of five cutting tools. The experiment is to select one part. Define a random variable for the experiment. Construct the probability distribution. Construct a cumulative probability distribution.

18. prob18 EXPECTED VALUE Discrete Random Variable E(X) = X 1 P(X 1 ) + …. + X n P(X n )

19. prob19 Example At a carnival, a game consists of rolling a fair die. You must play $4 to play this game. You roll one fair die, and win the amount showing (e.g... if you roll a one, you win one dollar.) If you were to play this game many times, what would be your expected winnings? Is this a fair game?

20. prob20 CUMULATIVE PROBABILITY FUNCTIONS For a discrete random variable X, the cumulative function is: F(X) = P(X < x) = f(z) for all z < x

21. prob21 PROBABILITY HISTOGRAMS

22. prob22 Variance of a Discrete Probability Distribution Var(X) =  [x - E(X)] 2 * f(x)

23. prob23 SOME SPECIAL DISCRETE RV’s  Binomial  Poisson  Geometric  Hypergeometric

24. prob24 BINOMIAL X = the number of successes in n independent Bernoulli trials of an experiment f(x) = n C x p x (1-p) n-x for x = 0,1,2….n f(x) = 0 otherwise

25. prob25 EXAMPLE A manufacturer claims only 10% of his machines require repair within one year. If 5 of 20 machines require repair, does this support or refute his claim??

26. prob26 POISSON DISTRIBUTION X = # of success in an interval of time, space, distance f(x) = e -  x /x! for x = 0,1,2,…... f(x) = 0 otherwise

27. prob27 EXAMPLES Examples of the Poisson number of messages arriving for routing through a switching center in a communications network number of imperfections in a bolt of cloth number of arrivals at a retail outlet

28. prob28 EXAMPLE of POISSON The inspection of tin plates produced by a continuous electrolytic process. Assume that the number of imperfections spotted per minute is 0.2.  Find the probability of no more than one imperfection in a minute.  Find the probability of one imperfection in 3 minutes.

29. prob29 GEOMETRIC DISTRIBUTION X = # of trials until the first success f(x) = p x (1-p) n-x for x = 0,1,2….n f(x) = 0 otherwise

30. prob30 Example of Geometric The probability that a measuring device will show excessive drift is 0.05. A series of devices is tested. What is the probability that the 6th device will show excessive drift? Find the probability of the 1 st drift on the 6th trail. P(X=1) = (0.05)(0.95) 5 = 0.039