Probability concepts

Chapter 5



  1. Definitions


    1. Probability is a value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.


    2. A experiment is a process that leads to the occurrence of one and only one of the several possible observations (outcomes).


    3. An outcome is a particular result of an experiment.


    4. An event is a collection of one or more outcomes of an experiment.


    5. Mutually exclusive: Events are mutually exclusive if the occurrence of one event means that none of the others can occur at the same time.


    6. Collectively Exhaustive: A list of events is collectively exhaustive if at least one of the events must occur when an experiment is conducted.


    7. Joint probability is a probability that measures the probability that two or more events will happen concurrently.


    8. Events are independent if the fact that one occurs does not effect the probability that the other will occur.


  2. Approaches to probability


    1. Classical probability is sometimes called a priori and is based on information know in advance of the experiment.


      1. Calculation of classical probability uses the equation:


P(A) = x/n



Where x = number of favorable outcomes

n = total number of possible outcomes



      1. Example: Assume a workforce of 50 people includes 20 women. If the names were placed in a hat and one was drawn, what is the probability it would be a woman?


        1. Answer: 20/50 = .4


      2. Most games of chance are based on classical probability.
    1. Empirical concept of probability is based on the relative frequencies of outcomes that occurred during past experiments.


      1. Formula

P(A) = x'/n



Where: x' = Number of times event x occurred in past observations

n = total number of observations

(The notation is not a standard notation)

      1. Example: During the past four hours, the number of cars passing the intersection of I 40 and US69 was 1500. Of the 1500, 140 were Fords. What is the probability that the next car will be a Ford?


        1. Answer:


P(Ford) = 140/1500 = .0933





    1. Subjective Probability is a probability assigned by an individual based on whatever information is available.


      1. Subjective probability does not involve a calculation in the same sense as classical and empirical probability.


      2. Example:


        1. What is the probability that XYZ Inc. will default on its loan?


        2. What is the probability that Podunk University will win its next basketball game?
  1. The Rules of Probability


    1. The special rule of addition is used when the outcomes of the experiment are mutually exclusive


      1. P(A or B) = P(A) + P(B)


      2. The special rule of addition can be extended to three or more mutually exclusive events


        1. P(A or B or C) = P(A) + P(B) + P(C)


    2. The complement rule


      1. P(A) + 1 - P(~A)


        1. The probability of A equals one minus the probability of "not A."
    3. General Rule of Addition is used when the events are not mutually exclusive.


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


        1. P(A and B) is an example of joint probability.


      2. Example problem: Using the data on the following contingency table, determine the probability that a randomly selected employee would be a male or from department A.


      3. Response:

P(M or B) = 16/35 + 8/35 - 3/35

= 21/35 = .6000





Dept. A Dept. B Dept. C Total
Female 5 10 4 19
Male 3 6 7 16
Total 8 16 11 35










    1. The Special Rule of Multiplication-requires that events be independent.


      1. Formula:


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



        1. Read P(A and B) as the probability of event A followed by event B, or that A and B occur at the same time.


      1. Example: XYZ Inc. gives a door prize to one worker each day. All names are placed in a hat each day and one is drawn. There are 19 women and 16 men. What is the probability that the name of a woman would be drawn the first two days the prize is given.


      2. Response:

P(W and W) = 19/35 * 19/35

= 361/1225

= .2947

    1. The General Rule of Multiplication


      1. Formula:


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



      1. Example: Assume two door prizes are given by XYZ Inc. on a given day and that the same person cannot win both. What is the probability that both prizes will be won by women.


      2. Response:


P(W and W) = 19/35 * 18/34



= 342/1190 = .2874