# Probability and Probability Distributions

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1 Chapter 1-3 Probability and Probability Distributions 1-3-1

2 Probability The relative frequency of occurrence of an event is known as its probability. It is the ratio of the number of favorable events to the total number of possible events

3 Example An inspector checks a sample of 1000 parts for conformance to standard. 40 do not meet standard. What is the probability of a defect? P(d) = 40/1000 =

4 Probability Definitions Two events are mutually exclusive if they have no outcomes in common or if when one occurs the other cannot. Two events are non-mutually exclusive when they share a common area of occurrence or can both happen at the same time

5 Addition Rule for Mutually Exclusive Events P(A or B) = P(A) + P(B) 1-3-5

6 Example A sample of 100 employees is selected at random from the company. Within this sample are 61 males and 39 females. Find the probability of Finding a male Finding a female Finding a male or a female 1-3-6

7 Solution Male: 61/100 =.61 Female: 39/100 =.39 Male or Female = = 1.00 (Addition Rule) 1-3-7

8 Replacement Sampling A sample or group of samples is returned to the population after sampling so that the probabilities associated with selecting additional items are not changed

9 Multiplication Rule for Independent Events The probability that one event will occur and that a second independent event will also occur requires the use of the multiplication rule. P(A and B) = P(A)P(B) 1-3-9

10 Example What is the probability of selecting two clubs in two draws from a deck of cards if each card is replaced in the deck after it is drawn? P(club) = 13/52 P(club and club) = (13/52)(13/52) = (169)/(2704) =

11 Probability Distributions Complete listing of all possible outcomes along with the likelihood that each will occur. Discrete distribution is one in which the observed characteristics fit into a finite number of categories. Continuous distribution is one in which the observed characteristic may take on any value within a given range

12 Practical Uses of Probability Distributions Some practical uses of probability distributions are: To calculate confidence intervals for parameters and to calculate critical regions for hypothesis tests. For univariate data, it is often useful to determine a reasonable distributional model for the data. Statistical intervals and hypothesis tests are often based on specific distributional assumptions. Before computing an interval or test based on a distributional assumption, we need to verify that the assumption is justified for the given data set. In this case, the distribution does not need to be the best-fitting distribution for the data, but an adequate enough model so that the statistical technique yields valid conclusions. Simulation studies with random numbers generated from using a specific probability distribution are often needed

13 Probability Distributions The values of a probability distribution must be numbers on the interval from 0 to 1. The sum of all the values of a probability distribution must be equal to

14 Sample Probability Distribution Dice Outcome Ways to Achieve Ways Total: Data Set Relative Frequency

15 Binomial Distribution A discrete probability distribution used in defining the probability of favorable occurrences in a sample. Often used to define the probability of finding defects in a sample. Each event must have a constant probability of occurrence and each must be independent

16 Binomial Probability Formula n! P( x) x!( n x)! p Mean np Stdev np( 1 p) x ( p) nx 1 P(x) is the probability of a favorable event x n is the sample size p is probability of single favorable event! Is the symbol for factorial x is the favorable event

17 Example A component supplied by a certain vendor has a 10 percent chance of being defective. Upon receipt of a lot a sample of 10 is selected at random. What is the probability the sample will contain 2 defects 2 or fewer defects

18 Solution Part a Because an item can be good or defective this is binomial. n = 10 p =.1 x = 2 P(x) = ((n!)/x!(n-x)!)(p) x (1-p) n-x P(2) = ((10!)/(2!(10-2)!))(.1) 2 (.90) 8 P(2) = (45)(.010)(.4305) =

19 Solution Part b The probability of 2 or fewer is the probability of 2 or 1 or 0. P(2) =.1937 P(1) = ((10!)/(1!9!))(.1) 1 (.9) 9 =.3874 P(0) = ((10!)/(0!10!))(.1) 0 (.9) 10 =.3487 P(2 or less) = P(2 or less) =

20 Probability Table Values of the binomial probability distribution are included in the appendix to this training book and are found in Table A

21 1-3-21

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23 SPCXL

24 1-3-24

25 Poisson Distribution The Poisson distribution is a discrete probability distribution used to determine the probability of x occurrences in a sample of n where the probability of a favorable event is constant, but relatively small. Often used when the probability is expressed as a rate

26 Poisson Probabilities P( x) Mean Stdev ( np) x x! np np e np P(x) is the probability of a favorable event x n is the sample size p is probability of single favorable event! Is the symbol for factorial x is the favorable event e =

27 Example Use the Poisson table (Table B) to answer the following: A textile manufacturer has kept records on the number of defects per yard of material. The probability has been.04. What is the probability that 20 yards of material will have 2 defects 3 or fewer defects

28 Excel

29 SPCXL

30 Normal Probability Distribution A continuous probability distribution used when there is a concentration of observations about the mean and equal likelihood that observations will occur above and below the mean

31 THE NORMAL LAW OF ERROR STANDS OUT IN THE EXPERIENCE OF MANKIND AS ONE OF THE BROADEST GENERALIZATIONS OF NATURAL PHILOSOPHY. IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES IN THE PHYSICAL AND SOCIAL SCIENCES AND IN MEDICINE, AGRICULTURE, AND ENGINEERING. IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION

32 Normal Probability Distribution The mean and standard deviation are calculated with standard procedures. Distribution is symmetrical about the mean. y e 1 x ( ) 2 s s

33 Standard Normal Curve Normal probability distribution with a mean of 0 and a standard deviation of 1. Values tabulated z x s x (Table C) Table accessed with z transform (typical format shown)

34 Example Sixteen samples of product weight were measured. They averaged 80 with a standard deviation of 6. What is the probability that a sample taken at random from future production will be a. Greater than 83? b. Less than 78?

35 Example A product has specs of and A sample of 25 has an average of and a standard deviation of.1 What is the probability that an item will be a. Greater than 16.05? b. Less than 16.00? c. Between and 16.04?

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38 Distribution of Sample Means z x - s n A population is normally distributed with a mean of 62 and a standard deviation of 3.5. What is the probability that a sample mean taken from a sample of size 16 will be larger than 60?

39 Chi Square Distribution The chi-square distribution results when independent variables with standard normal distributions are squared and summed. The formula for the probability density function of the chi-square distribution is f ( x) e 2 x 2 n 2 x n 1 2 n G( ) 2 for x 0 where n is the shape parameter and G is the gamma function. The formula for the gamma function is G( a) t 0 a1 e l dt

40 1-3-40

41 F Distribution The F distribution is the ratio of two chi-square distributions with degrees of freedom n 1 and n 2, respectively, where each chi-square has first been divided by its degrees of freedom. The formula for the probability density function of the F distribution is f n1 n 2 n1 G( )( ) 2 n n1 2 2 ( x) n1n 2 n1 n 2 n1x 2 G( ) G( )(1 ) 2 2 n where n 1 and n 2 are the shape parameters and G is the gamma function. 2 x n

42 1-3-42

43 Student t Distribution The t distribution is used instead of the normal distribution whenever the standard deviation is estimated. The t distribution has relatively more scores in its tails than does the normal distribution. As the degrees of freedom increases, the t distribution approaches the normal distribution

44 Student t Distribution The formula for the probability density function of the t distribution is f ( x) 2 ( n 1) x 2 (1 ) n b (0.5,0.5n ) where b is the beta function and n is a positive integer shape parameter. The formula for the beta function is n b 1 b 1 (, b) t (1 t) dt

45 1-3-45

46 Hypergeometric Distribution A distribution closely related to the binomial that has a wide applicability in acceptance sampling. x is the number of favorable events N is the lot size n is the sample size D is the number of favorable events in the population P(x) D x N n N n D x [ ] Notation means combinations

47 Which Distribution to Use? The question continues to arise regarding which probability distribution should be used when. The following conditions, originally proposed by Burr in Elementary Statistical Quality Control address this issue. Conditions for a Binomial Distribution Constant probability of a defective on each draw of a piece is established. Results on drawings are independent of preceding results. n draws of a piece are taken. Number of defectives in the n draws are counted. Either/Or

48 Conditions for a Poisson Distribution Samples provide equal areas of opportunity for defects (the same size of unit, subassembly, length, area, or quantity.) Defects occur randomly and independently of each other. The average number remains constant. The possible number of defects is far greater than the average number of defects. Rate

49 Properties of a Standard Normal Distribution The total area lying between the curve and the horizontal z axis is 1, representing a total probability of 1. The height of the curve represents relative frequency. The relative frequency is greatest at z = 0. Relative frequencies steadily decrease as z moves away from 0 in either direction. The relative frequency rapidly approaches 0 in both directions. The curve is symmetrical around z = 0. Desired probabilities for ranges of z may be found from published tables of normal curve areas

50 Some Common Applications of Distributions Distribution Uses Hypergeometric Sampling Binomial Go-No Go Sampling Confidence Intervals - Proportions Control Charts-Proportion Defective Poisson Sampling Queueing - Arrival Rates Control Charts-Defects per Unit Normal Process Capability Tests of Hypothesis-Means, Correlation Coefficients Confidence Intervals - Means Student t Tests of Hypothesis Means Confidence Intervals - Means Chi Square Goodness of Fit Testing Confidence Intervals Standard Deviations F Tests of Hypothesis Standard Deviations Analysis of Variance Exponential Reliability Queueing - Service Rates Weibull Reliability

51 Descriptive Statistics Distribution Mean Standard Deviation Binomial np np(1 - p) Poisson np np Normal Student t x n x n (x n (x n x ) x ) 2 2 Exponential x n x n

52 Practice Problems In problems 1-3, use a binomially distributed sample with p =.2 and n = 14 to determine the probability of each event. 1. x is equal to x is less than x is greater than x is between 2 and Calculate the mean and the standard deviation for a binomially distributed sample with p =.2 and n =

53 Practice Problems In problems 6-10 use a binomially distributed sample with p =.6 and n = 10 to determine the probability of each event. 6. x is equal to x is greater than x is less than x is greater than x is between 0 and

54 Practice Problems In problems use a binomially distributed sample with p =.45 and n = 20 to determine the probability of each event. 11. x is greater than x is less than x is equal to x is between 2 and 7. In problems given that a sample follows the Poisson distribution and has a mean of.95, determine the requested value. 15. The standard deviation. 16. The probability that x is greater than The probability that x is less than The probability that x is between 1 and

55 Practice Problems 19. Given a normal probability distribution with a mean of 79 and a standard deviation of 4, at what value would we expect to find at least as large as 70? 20. Given a normal probability distribution with a mean of 20 and a standard deviation of.5. If we want to have 99% of all the process output between symmetrical specifications what should those specifications be? 21. What z value corresponds to having a probability between the point and minus infinity of.025? 22. What z value corresponds to have a probaiblity between the point and positive infinity of.100?

56 Practice Problems In problems 23-32, determine the probability of each event for a normally distributed process with a mean of 120 and a standard deviation of x is greater than x is less than x is greater than x is less than x is greater than x is between 90 and x is between 125 and x is between 128 and x is between 116 and x is equal to

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