Section 7

Fundamentals of

Probability

2

Probability

• Probability

– likelihood– chance– tendency– trend

3

Probability

• The chance that something will happen.

– It will rain tomorrow.

– I will play golf tomorrow.

– I will receive an “A” in this course

4

Probability

• Coin toss

• Dice

• Cards

5

Probability

• Event

– Is a collection of outcomes.

• I.e. with a six-sided die you would have 6 possible outcomes.

6

Probability

• P(A) = NA/N

• Where P(A) = probability of an event A occurring to 3 decimal places

• NA = number of successful outcomes of event A

• N = total number of possible outcomes

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Probability

• The probability using known outcomes is the true probability.

• The one calculated using experimental outcomes is different due to the chance factor.

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Probability

• The previous definition is useful for finite situations where NA, the number of successful outcomes and N, total number of outcomes are known or must be found experimentally.

9

Probability

• For an infinite situation, where N = infinity, the definition would always lead to a probability of zero.

• In the infinite situation the probability of an event occurring is proportional to the population distribution.

10

Probability

• Theorems of Probability– Theorem 1

• Probability is expressed as a number between 1.000 and 0, where a value of 1.000 is a certainty that an event will occur and a value of 0 is a certainty that an event will not occur.

– Theorem 2• If P(A) is the probability that event A will occur,

then the probability that A will not occur.

11

Probability

• Theorem applicability (Figure 7-2)

– Probability of only one event occurring, then use theorem 3 or 4 depending on mutual exclusivity.

– 2 or more events desired then use theorem 6 or 7 depending on if they are independent or not.

12

Probability

• Theorem 3

– If A and B are two mutually exclusive events, then the probability that either event A or event B will occur is the sum of their respective probabilities.

– P(A or B) = P(A) + P(B)

13

Probability

• Mutually Exclusive– Means that the occurrence of one event makes

the other event impossible.– Whenever an “or” is verbalized, it is usually

addition.– Theorem 3 is referred to as the “additive law of

probability.”

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Probability

• Theorem 4– If event A and event B are not mutually

exclusive events, then the probability of either event A or event B or both is given by:

• P(A or B or both) = P(A) + P(B) - P(both)

– Events that are not mutually exclusive have some outcomes in common.

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Probability

• Theorem 5

– The sum of the probabilities of the events of a situation is equal to 1.000

• P(A) + P(B) + . . . + P(N) = 1.000

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Probability

• Theorem 6

– If A and B are independent events, then the probability of both A and B occurring is the product of their respective probabilities.

• P(A and B) = P(A) X P(B)

– An independent event is one where its occurrence has no influence on the probability of the other event or events.

– Referred to as the “Multiplicative Law of Probabilities.

– When an “and” is verbalized, the mathematical operation is

multiplication.

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Probability

• Theorem 7– If A and B are dependent events, the probability

of both A and B occurring is the product of the probability of A and the probability that if A occurred, then B will occur also.

• P(A and B) = P(A) X P(B\A)

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Probability

• The symbol P(B/A) is defined as the probability of event B provided that event A has occurred.

• A dependent event is one whose occurrence influences the probability of the other event or events.

• Referred to as the “Conditional theorem”

19

Probability

• Counting of Events

– 3 counting techniques that are used in the computation of probabilities.

• Simple multiplication

• Permutations

• Combinations

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Probability

• Simple multiplication

– If an event A can happen in any of a ways or outcomes and, after it has occurred, another event B can happen in b ways or outcomes, the number of ways that both events can happen is ab.

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Probability

• Permutations

– Is an ordered arrangement of a set of objects.

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Probability

• Combinations

– If the way the objects are ordered is unimportant, then we have a combination.

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Probability

• Discrete Probability Distributions

– If specific values such as integers are used, then the probability distribution is discrete.

• Hypergeometric

• Binomial

• Poisson

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Probability

• Hypergeometric Probability Distribution– Occurs when the population is finite and the

random sample is taken without replacement.

– Formula is made from (3) combinations.• Total combinations

• nonconforming combinations

• conforming combinations

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Probability

• The numerator– Is the ways or outcomes of obtaining

nonconforming units times the ways or outcomes of obtaining conforming units.

• The denominator– Is the total possible ways or outcomes

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Probability

• Binomial Probability Distribution– Is applicable to discrete probability problems

that have• an infinite number of items or

• that have a steady stream of items coming from a work center.

– Is applied to problems that have attributes• conforming or nonconforming

• pass or fail

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Probability

• Binomial is used for– infinite situations– requires that there only be two outcomes

• conforming or nonconforming

– that the probability of each outcome does not change

– trials are to be independent

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Probability

• Poisson Distribution Distribution– Named after Simeon Poisson, 1837– Applicable to situations that involve:

• observations per unit of time

• observations per unit of amount

– There are many equal opportunities for the occurrence of an event

– Is the basis for attribute control charts and for acceptance sampling

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Probability

• Continuous Probability Distributions– Normal Probability Distribution

• Measurable data, meters, kilograms, ohms

– Exponential Probability Distribution• Used in reliability studies with constant failure rates

– Weibull• used when the time to failure is not constant

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Probability

• Distribution Interrelationship

– Use Poisson whenever appropriate• Can be easily calculated

– Use Hypergeometric for finite lots

– Use the Binomial for infinite situations or when there is a steady stream of product