Basic Probability Concepts

Question 1

Basic Probability

Answer 1

Basic Six Sigma Probability terms like independence, mutually exclusive, compound events, and more are necessary foundations for statistical analysis.

Probability is the ratio of number of favorable outcomes to the total number of possible outcomes. Probabilities are usually shown in fractions or decimals. The probability ALWAYS lies between 0 and 1. An event is one or more outcomes in an experiment. The probability of an event E indicates how likely that event is to occur.

Probability of an event (E) = number of favorable outcomes/number of possible outcomes

• Additive law
• Multiplication law
• Compound Event

Question 2

Additive Law

Answer 2

• Aka addition rule
• The probability of the union of two events. There are two scenarios in the additive law:
  
  • When two events are not mutually exclusive:
    
    • When two events A and B are not mutually exclusive, the probability of A and B will occur is the sum of the two events probabilities and subtract both probability of A and B will occur (intersection), the formula can summarize the same:
      
      • P (A U B) = P(A) + P(B) – P(A Ո B)
  • When two events are mutually exclusive:
    
    • When two events A and B are mutually exclusive, the probability of A and B will occur is the sum of the two events’ probabilities, the formula can summarize same:
      
      • P (A U B) = P(A) + P(B)

Question 3

Mutually Exclusive

Answer 3

Statistical term describing two or more events that cannot happen simultaneously.

If they are** mutually exclusive they **cannot happen simultaneously

If they are not mutually exclusive, they can happen simultaneously

Question 4

When Would you Use the Additive Law in a Six Sigma Project?

Answer 4

Additive law is essential in probability. Additive law tells us with a way to calculate the probability an event “A” or the probability of “B.” The use of the additive law is dependent upon whether event A and event B are mutually exclusive or not.

Example Attached

Question 5

Multiplication Law

Answer 5

• A method to find the probability of events occurring at the same time. There are two scenarios in the multiplication law:
  
  • When events are independent
    
    • When two independent events A and B are occurring from the same sample space, then the probability of two events occurring at the same is equal to the probability of A occurs times the probability of B occurs
      
      • P (A Ո B) = P(A) * P(B)
• When events are dependent
  
  • When two dependent events A and B are occurring from the same sample space, then the probability of two events occurring at the same is equal to the probability of A occurs times the probability of B occurs, given that A has already occurred. This is conditional probability
    
    • P (A Ո B) = P(A) * P(B|A)

Conditional Probability: If events A and B are dependent, the probability of A influences the probability of B

All Multiplication Law:

P(A&B) = P(A)*P(B)

P(A&B)= P(A)*P(B|A)

P(A& B)= P(B)*P(A)

P(A&B)= P(B)*P(A|B)

Question 6

Compound Event

Answer 6

• It is an event that has more than one possible outcome of an experiment. In other words, compound events are formed by a composition of two or more events.
• When would you guys compound events?
  
  • Used when the outcome may have different probabilities but they are all equally possible. Events that are chained together in a row.

Example Attached

Question 7

Independent Event

Answer 7

• Events can be independent events when the outcome of one event does not influence another event’s outcome.
• When would you use an independent event in a six sigma project?
  
  • Independent events are used in six sigma projects where one event does not connect with another event’s chance of occurring. Like car mileage does not depend on the color of the car.

Example Attached

P(A|B)=P(A)

Question 8

Hypothesis Testing

Answer 8

• Key procedure in inferential statistics used to make statistical decisions using experimental data.
• It is basically an assumption that we make about the population parameter.
• When using hypothesis testing we create:
  
  • null hypothesis (H0): the assumption that the experimental results are due to chance alone, nothing (from 6M) influenced out results.
  • alternative hypotheses (Ha): we expected to find a particular outcome.

Question 9

Determine the Process Capability

Answer 9

• Process Capability Analysis
  
  • Tells us how well a process meets a set of specification limits based on a sample of data taken from a process.
  • The process capability study helps to establish the process baseline and measure the future state performance.
    
    • Revisit the operational definitions and specify what are defects and which are opportunities.
• Calculate the baseline process sigma
  
  • The value in making a sigma calculation is that it abstracts your level of quality enough so that you can compare levels of quality across different fields (and different distributions.) In other words, the sigma value (or even DPMO) is a universal metric, that can help yourself with the industry benchmark/competitors.
    
    • Baseline Sigma for Discrete Data
    • Baseline Sigma for Continuous Data

Question 10

Baseline Sigma for Discrete Data aka Attribute Data

Answer 10

• Calculate the process capability is through the number of defects per opportunity.
• The acceptable number to achieve sex sigma is 3.4 Defects Per Million Opportunities (DPMO)
  
  • DPO= Defects/(Units*Opportunities)
  • DPMO=(Defects/Units*Opportunities*Total 1,000,000
  • Yield=1-DPO(It is the ability of the process to produce defect free units)

Question 11

Baseline Sigma for Continuous Data

Answer 11

• Process Capability is the determination of the adequacy of the process with respect to the customers needs.
  
  • Process capability compares the output of an in-control process to the specification limits.
  • Cp and Cpk are considered short-term potential capability measures for a process.

Question 12

Cpk

Answer 12

• Cpk is a measure to show how many standard deviations the specification limits are from the center of the process.
  
  • C_{plower =} (Process Mean - LSL)/(3*Standard Deviation)
  • C_pupper = (USL - Process Mean)/(3*Standard Deviation)
  • Cpk is the smallest value of the Cpl or Cpu
    
    • C_pk = Min (C_pl, C_pu)

Question 13

Six Sigma Derives…

Answer 13

• Six Sigma derives from the normal or bell curve in statistics, where each interval indicates one sigma or one standard deviation
• Sigma is a statistical term that refers to the standard deviation of a process about its name
  
  • In normally distributed process, 99.73% of measurement will fall within ±3σ and 9.99932% will fall within ±4.5σ.

Question 14

Mutually Exclusive Events

Answer 14

• Mutually exclusive events are things that cannot both be true.
• They cannot occur at the same time.

When would we use mutually exclusivity in a six sigma project?

-Mutually exclusive events used in six sigma projects when one event occurs it prevents the second from happening

Coin Heads/Tails and dice example attached

Question 15

Complementary Rule of Probability

Answer 15

• The comment of A is represented by A’. The complement (A’) includes all outcomes in the sample space that are not the outcomes of event A.
  
  • In other words, it consists of all outcomes in which event A does not occur
• An event (A) and its complement (A’) are mutually exclusive, which means either event (A) or its complement (A’) will occur, not both at a time.
  
  • The complement rule is the sum of probability of an event (A), and its complement (A’) is 1.
    
    • P(A) +P(A’)=1
  • Example: When tossing a coin, if the event (A) is tails, then the complement (A’) is heads. If you are throwing a die and the event (A) is even {2,4,6} then the complement (A’) is {1,3,5}

Question 16

What are Independent Events?

Answer 16

• Events can be independent events when the outcome of the one event does not influence another event’s outcome.
  
  • In other words, each event is NOT affected by others
• Whereas the dependent event is the probability of one event occurring that influences another event

Question 17

Probability Question

Answer 17

John has a deck of cards. He has randomly drawn a card and placed on table without showing you what is it. Now he is going to draw a second card. What is the probability that is is going to be one of the hearts?

Question 18

What is the probability of selecting a Queen of any suit or any Heart in a standard 52 card deck?

Answer 18

Question 19

See Attached Example

Answer 19

Question 20

Answer 20

Question 21

Answer 21

Question 22

Answer 22

Question 23

Cpk Example

Answer 23

Question 24

Example Including Runout Tolerance

Answer 24

Question 25

Short Term Process Capability Example

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28