 A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences P(n) = probability of n occurrences p= proportion success (what you are looking for) q= proportion failures (what you are not looking for) Example: If a fair coin is tossed, what is the probability of a head occurring?

 You are rolling a fair die. A defective product is a roll of a 1. What are the odds that you will find a defective product?

 Since probabilities are a ratio, or expressed as percentage, then: 0 or 0%  ”impossible event” 1 or 100%  “sure thing” Example: You are rolling a fair six-sided die. What are the odds that you will roll a 7? Not a 7?

 When more than one success can occur, we describe that as “or” ◦ sum of the individual probabilities (+)  When more than one success needs to occur, we describe that as “and” ◦ product of the individual probabilities (x)

 You are rolling a fair die. What are the odds that you will roll a 2 or a 4? A 2 and a 4?

 This equation is used when the events are “mutually exclusive” – meaning they do not occur at the same time  If the successful event can occur more than once, you must use an adjusted equation Example: Draw a King or a Queen

 You have a standard deck of cards. What are the odds that you draw a 3 or club?

 This equation is used when the events are “independent” – meaning they do not affect each other  If the successful event can affect each other, then you must use an adjusted equation Example: Flip of a coin

10  You have a standard deck of cards. What are the odds that you will draw four aces without replacement?

11  Number of ways is a listing of possible successes  Permutation, PN,n, P(n,r), nPr, the number of arrangements when order is a concern – “think word”  Combination, CN,n,, nCr, the number of arrangements when order is not a concern

12  The product of a number and all counting numbers descending from it to 1 6! = 6x5x4x3x2x1=720 Note: 0!=1

13  How many 3 letter arrangements can be found from the word C A T? How about 2 letter arrangements?  Three lottery numbers are drawn from a total of 50. How many arrangements can be expected?

14  How many 3 letter groupings can be found from the word C A T?  Three lottery numbers are drawn from a total of 50. How many combinations can be expected?

15  Refers to the probability of two possible outcomes, success (s) and failure (f)  Example: Let’s look at the possibilities of flipping coins *See table 5.1 pg flip = H(s) or T(f) 2 flips = HH or HT or TH or TT Etc. Calculated by:

16  A single die is tossed five times. Find the probability of rolling a four, three times.

17  Refers to the probability model that can be used for non- replacement sampling  Uses combinations and the basic probability formula

18  A manufacturer has received 12 parts from a supplier, 10 are good. If a sample of 4 are taken, find the probability of picking 3 good parts.

19 Each unit of measure is a numerical value on a continuous scale Size Pieces vary from each other Variation common and special causes But they form a pattern that, if stable, is called a normal distribution Histogram or Frequency Distribution Normal Distribution

20 There are three terms used to describe distributions 3. Location Mean 1. Shape Bell 2. Spread Standard Deviation

21  Symmetrical, Bell-Shaped  Extends from Minus Infinity to Plus Infinity  Two Parameters ◦ Mean or Average ( ) ◦ Standard Deviation ( )  Space under the entire curve is 100% of the data  Mean, median and mode are the same

22 50% -1  -2  -3  +1  +2  +3  0   ≈ 68% 1   99.73% 3   ≈95% 2 z value = distance from the mean measured in standard deviations

23  Normal Curve theory tells us that the probability of a defect is smallest if you ◦ stabilize the process (control) ◦ make sigma as small as possible (reduce variation) ◦ get Xbar as close to target as possible (center) So… For SPC we first want to stabilize the process, second we will reduce variation and last thing is to center the process.

24  Specifies the areas under the normal curve  Represents the distance from the center measured in standard deviations  Values found on the normal table pg Population Sample Remember when we talked about  3  ? The 3 is the z value.

25  The known average human height is 5’8” tall with a standard deviation of 5 inches. What are the z values for 6’2” and 4’8”? A positive value indicates a z value to the right of the mean and a negative indicates a z value to the left of the mean.

26  From our answers from the last exercise, what is the values for: ◦ P(Area > 6’2”)? ◦ P(Area < 4’8”)? ◦ P(4’8”< Area < 6’2”)? ◦ Prove area under the normal curve at  1s,  2s,  3s?

27  Suppose the HR department decided to only hire people between the heights of 6’8” and 4’9” tall. What percentage of the population, based on our sample, would we not be able to hire?

28  States that any distribution of sample means from a large population approaches the normal distribution as n increases to infinity  If you chart the values, the values will have less variation than the individual measurements  Standard deviation is expressed as:

29  Let’s look at an example of how the central limit theorem works