Business Statistics: Chapter 5; A Survey Of Probability Concepts

Probability

    Probability: Value between 0 and 1 that describes relative possibility (chance or likelihood) an event will occur.

• Possibility close to 0: indicate chance of event happening is low.

• Possibility close to 1: indicate chance of event happening is high.

    Experiment: Leads to occurrence of one and only one of several potential results.

    Outcome: Specific result of an experiment.

    Event: Collection of one or more outcomes of an experiment.

Principles of Counting

    Three Principles of Counting: Help determine the number of all possible outcomes.

    Multiplication Formula:

        Total Number of Arrangements= (m)(n)

        Where m= n= number of possibilities of an event and n= number of possibilities of another event

    Permutation Formula: any arrangement of r objects selected from a single group of n possible objects.

        nPr= n! / (n-r)!

        Where n= total number of objects and r= number of objects selected. ! is factorial, meaning that if its 5!, it is (5)(4)(3)(2)(1).

    Combination Formula: An event of outcomes when order of outcomes doesn’t matter. Count the number of r objects combinations from a set of n objects.

        nCr= n!/ r! (n-r)!

        Where n= total number of objects and r= number of objects selected.

Assigning Probabilities

    Classical Probability: based on assumption that all outcomes of an experiment are equally likely. Based on equally likely outcomes.

    Probability of an Event= # Favorable Outcomes /

                                              # Possible Outcomes

    Empirical Probability: Based on the observation, counting, recording of experimental outcomes. Based on relative frequencies.

    Empirical Probability= # Times Event Occurs /

                                            # Total Observations

    Subjective Probability: Probability of an event that is assigned by whatever information available.

Rules of Addition

    Special Rule of Addition: Use to find probabilities of events made up of A or B, when events are mutually exclusive (when one event occurs, no other events can occur at the same time.)

        P (A or B) = P (A) + P(B)

        or, Possibility of A or B = Possibility of A + Possibility of B

    Complement Rule: simplest and most important rule of probability. Used to find probability of an event occurring by subtracting probability of the event not occurring from 1.

        P (A) = 1 - P (A)

    General Rule of Addition: used to find probability of either two events happening.

        P (A or B) = P (A) + P (B) - P ( A and B)

    Joint Probability: probability of two events occurring at the same time.

Rules of Multiplication

    Special Rule of Multiplication: requires two events to be independent, occurrence of one event having no effect on probability of another event.

        P (A and B) = P (A) * P (B)

    General Rule of Multiplication: When two events are dependent, using conditional probability, meaning probability of event occurring given another has occurred.

        P (A and B) = P (A) * P (B given A)

            Example: 12 shirts, 9 white and 4 blue. To find probability of two white shirts being selected, 9/12 × 8/11.

Contingency Tables

    Contingency Table: used to classify sample observations according to two or more categories/classes.

    Tree Diagrams: visual to help organize and calculate probabilities.

Excel Functions

    Counting Principles using Permut and Combin:

• For permutation, use =PERMUT(# total objects, # chosen)

• For combination, use =COMBIN(# total objects, # in each combination)