PROBABILITY DISTRIBUTIONS

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BASIC PROBABILITY


Purpose

To define basic probability theory and concepts that are helpful to understanding Six Sigma tools and their application.


Anatomy

basic probability theory

Reference: Juran’s Quality Control Handbook

Terminology

A. Random Experiment - Any process that involves chance and that is likely to lead to one or more results. A random experiment exhibits the following characteristics:

  • The result cannot be predicted with certainty;
  • The set of all possible results can be described before the experiment;
  • The experiment can be repeated at will under the same conditions.

B. Sample Space - The set of all possible outcomes from a random experiment.

C. Event - Any possible subset of the sample space.

D. Probability - A number that is calculated from a sample and that indicates the likelihood of an event occurring in a population. It can be thought of as a number that indicates the proportion of times an event would occur if an experiment is repeated a very large number of times. This number ranges from 0.0 (impossibility of occurrence) to 1.0 (certainty of occurrence), and the sum of the probability of all possible outcomes is equal to 1.0. When all events are equally likely to occur the probability of an event (A) is defined as:

calculating probability

THEOREMS


Purpose

To describe three basic theorems helpful in understanding probabilities.

Anatomy

probability theorems

Reference: Juran’s Quality Control Handbook

Terminology

A. Theorem 1 - If P(A) represents the probability that event A will occur, then the probability that event A will not occur, P(A’) is equivalent to P(A’) = 1 - P(A).

B. Theorem 2 - If A and B are two events that can occur simultaneously, then the probability that event A or event B will occur is P(A È B) = P(A) + P(B) - P(A Ç B).

Note: If A and B are mutually exclusive, then the probability of either A or B is
P(A
È B) = P(A) + P(B).

C. Theorem 3 - If A and B are two independent events, then the probability that events A and B occur simultaneously is P(A Ç B) = P(A) x P(B).

If the occurrence of A influences the probability that event B will occur, then the probability that event A and event B will occur simultaneously is P(A Ç B) = P(A) x P(B|A). P(B|A) is read “probability of B, knowing that A has occurred”

SETS AND EVENTS


Purpose

To describe operations involving sets. These descriptions are useful in understanding calculation of probabilities which are an essential tool in Six Sigma.

Anatomy

event operations

Reference: Juran’s Quality Control Handbook

Terminology

Composite event - A combination of various events. Each rectangle represents a set of events. For example, if we toss three coins and we are interested in the event: “More heads that tails” the following combinations (out of all eight possible combinations) represent a composite event: HHH, HHT, THH, HTH. The probability of a composite event is the sum of the probability of all simple events that comprise the composite event. P(more heads than tails) = P(HHH)+ P(HHT)+ P(THH)+ P(HTH)

A. Intersection A Ç B - The event (A Ç B) occurs only if both A and B occur.

B. Union A È B - The event (A È B) occurs if A or B or both A and B occur.

C. Negation or opposite of A (A’) - This event occurs if and only if A does not occur.

D. Inclusion or Implication A Í B - Read “A includes B”, is the event whereby every time A occurs, B occurs as well.

E. Events incompatible or mutually exclusive (A Ç B = Æ ) - This event occurs if events A and B cannot occur at the same time. In other words, if there is no element that is in both A and B. The intersection of A and B is an empty set (represented by Æ ).

CONDITIONAL PROBABILITY


Purpose

To calculate the probability of an event in relation to a second event instead of in relation to the sample space, e.g.: the probability of occurrence of event B knowing that event A has occurred.

Anatomy

conditional probability

Reference: Juran’s Quality Control Handbook

Terminology

A. Two successive events (Drawing two kings from an unmarked deck of cards. The following formula allows us to find the probability of occurrence of B knowing that A has occurred and the probability of A knowing that B has occurred.

probability distibution

The Multiplication Theorem allows us to calculate the probability of occurrence of two successive events:

probability

Multiplication Theorem - For two events with non-zero probability, the probability that the two events A and B will occur simultaneously is equal to the product of the probability of A multiplied by the probability of B, knowing that event A has occurred.

CONTINUOUS PROBABILITY DISTRIBUTION


Purpose

A Probability Distribution is a graph, table or formula used to assign probabilities to all of the possible outcomes or values of a characteristic measured in an experiment.

Anatomy

probability distribution

Reference: Juran’s Quality Control Handbook

Terminology

A. Continuous Probability Distribution - The characteristic measured can take on any value subject only to the precision of the measuring instrument (length, time, money, distance, etc.).

B. The sum of all probabilities is equal to 100% (i.e. the area under the curve equals one). Examples of continuous distributions include the Normal, Log normal, Exponential, Weibull, t, F, Chi-square, etc.

DISCRETE PROBABILITY DISTRIBUTION


Purpose

A Probability Distribution is a graph, table or formula used to assign probabilities to all of the possible outcomes or values of a characteristic measured in an experiment.

Anatomy

probability distribution

Reference: Juran’s Quality Control Handbook

Terminology

A. Discrete Probability Distribution - Graph describing the probabilities of occurrence of all possible results. In a discrete probability distribution the characteristic measured can only take on discrete values, i.e.: 0, 1, 2, 3, etc.

B. The sum of all probabilities is equal to 100%

Note: The sum of the probabilities shown above equals 94.4%. due to rounding. However the calculation is correct and the sum of the probabilities equals 100%. Examples of discrete distributions include the Poisson, Binomial, Negative Binomial, etc.

SAMPLE AND POPULATION


Purpose

To define basic probability theory and concepts helpful to understand Six Sigma tools and their application.

Anatomy

sample and population

Reference: Juran’s Quality Control Handbook

Terminology

A. Sample - A limited number of measurements taken from a larger source.

B. Population or Universe - A large source of measurements from which a sample is taken. A population may physically exist, or it may only be a concept, such as all experiments which might be run.

Reference: Juran’s Quality Control Handbook

Terminology

A. Sample - A limited number of measurements taken from a larger source.

B. Population or Universe - A large source of measurements from which a sample is taken. A population may physically exist, or it may only be a concept, such as all experiments which might be run.

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