Fundamentals of Probability Section 7 Fundamentals of Probability

Probability Probability likelihood chance tendency trend

Probability The chance that something will happen. It will rain tomorrow. I will play golf tomorrow. I will receive an “A” in this course

Probability Coin toss Dice Cards

Probability Event Is a collection of outcomes. I.e. with a six-sided die you would have 6 possible outcomes.

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

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

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.

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.

Probability Theorems of Probability Theorem 1 Theorem 2 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.

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.

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)

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.”

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.

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

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.

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)

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”

Probability Counting of Events 3 counting techniques that are used in the computation of probabilities. Simple multiplication Permutations Combinations

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.

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

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

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

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

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

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

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

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

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

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