1. Fundamentals of Probability
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

7. Probability
The probability using known outcomes is the true probability.
The one calculated using experimental outcomes is different due to the chance factor.

8. 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 Theorem 2
Probability is expressed as a number between and 0, where a value of 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.”

14. 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.

15. 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

16. 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.

17. 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)

18. 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

20. 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.

21. Probability Permutations
Is an ordered arrangement of a set of objects.

22. Probability Combinations
If the way the objects are ordered is unimportant, then we have a combination.

23. Probability Discrete Probability Distributions
If specific values such as integers are used, then the probability distribution is discrete.
Hypergeometric
Binomial
Poisson

24. 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

25. Probability The numerator The denominator
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

26. 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

27. 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

28. 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

29. 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

30. 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