Some Basic Concepts Probability

Random Experiment
A random experiment is a process leading to two or more possible outcomes, with uncertainty as to which outcome will occur.

Examples of random experiments include the following:

1.    A coin is tossed and the outcome is either a head or a tail.
2.    A customer enters a store and either purchases a shirt or does not.
3.    The daily change in an index of stock market prices is observed.
4.    A six sided die is rolled.

In each of the random experiments listed we can specify the possible outcomes, defined as basic outcomes. For example, a customer either purchases a shirt or not.

Sample space

The possible outcomes of a random experiment are called the basic outcomes, and the set of all basic outcomes is called a sample space. The symbol S will be used to denote the sample space. 

Example: What is the sample space for the roll of a single six-sided die?
Solution: The basic outcomes are the six possible face numbers, and the sample space is

                     S = {1, 2, 3, 4, 5, 6}

The sample space contains six basic outcomes. No two outcomes can occur together, and one of the six must occur.

Example: An investor follows the Dow-Jones Industrial index. What are the possible basic outcomes at the close of the trading day?
Solution: The sample space for this experiment is

             S = [{1. The index will be higher than at yesterday’s close},
                     {2. The index will not be higher than at yesterday’s close}]

One of these two outcomes must occur. They cannot occur simultaneously. Thus, these two outcomes constitute a sample space.

Event

An event, E, is any subset of basic outcomes from the sample space. An event occurs if the random experiment results in one of its constituent basic outcomes. The null event represents the absence of a basic outcome and is denoted by.

Example: Suppose a coin is tossed twice. Let H and T denote the head  and tail of the coin respectively. Then the sample space  of the experiment is
                 S = {HH, HT, TH, TT}
 Let A be the event of head of the first coin, then A will contain the sample points A = {HH, HT}.

Mutually Exclusive Events

If A and B be two events, then they are said to be mutually exclusive if . That is, two events are said to be mutually exclusive if they have no common points.
Example: Suppose a coin is tossed twice. Let H and T denote the head  and tail of the coin respectively. Then the sample space of the experiment is
                 S = {HH, HT, TH, TT}
Let A be the event of head of the first coin and B be the event of tail of the first coin, then A and B  will contain the sample points A = {HH, HT} and B = {TH, TT}. Since A and B have no point in common, A and B are said to be mutually exclusive.

Union

Let A and B be two events in the sample space, S. Their union, denoted A U B  , is the set of all basic outcomes in S that belong to at least one of these two events. Hence, the union A U B occurs if and only if either A or B or both occur.

More generally, given the K events E1, E2,……….,Ek, their union,  is the set of all basic outcomes belonging to at least one of these K events. 

Collectively Exhaustive

If the union of several events covers the entire sample space, S, we say that these events are mutually collectively exhaustive. Since every basic outcome is in S, it follows that every outcome of the random experiment will be in at least one of these events. Given the K events E1, E2,…………,Ek in the sample space, S, if =S , these K events are said to be collectively exhaustive. For example, if a die is thrown, the events “Result is at least 3” and “Result is at most 5” are together collectively exhaustive.

Complement

Let A be an event in the sample space, S. The set of basic outcomes of a random experiment belonging to S but not to A is called the complement of A and is denoted by

Example: A die is rolled. Let A be the event “Number resulting is even” and B the even “Number resulting is at least 4.” Then

                   A = {2, 4, 6}  and  B = {4, 5, 6}.
Find the complement of each event, the intersection and the union of A and B, and the intersection of  and B.
Solution: The complements of these events are, respectively

The intersection of A and B is the event “Number resulting is either even or at least 4, or both” and so

The union of A and B is the event “Number resulting is either even or at least 4, or both” and so

Note also that the events A and are mutually exclusive, since their intersection is the empty set, and collectively exhaustive, because their union is the sample space S; that is,

Example: Roll of a single die (Unions, Intersections, and Complements)
A die is rolled. Let A be the event “Number resulting is even” and B the event “Number resulting is at least 4.” Then
                    A = {2, 4, 6}       and B = {4, 5, 6}

Fine the complement of each event, the intersection and the union of A and B, and the intersection of   and B.
Solution: The complements of these events are, respectively,

                = {1, 3, 5}   and   = {1, 2, 3}

The intersection of A and B is the event “Number resulting is both even and at least 4” and so

The union of A and B is the event “Number resulting is either even or at least 4, or both” and so

Note also that the events A and  are mutually exclusive, since their intersection is the empty set, and collectively exhaustive, because their union is the sample space S; that is,

 Example: Dow-Jones Industrial Average (Unions, Intersections and Complements).
We will designate four basic outcomes for the Dow-Jones Industrial average over 2 consecutive days:
             Q1 : Dow-Jones average rises on both days.
             Q2 : Dow-Jones average rises on the first day but does not rise on the
                     second day.
             Q3 : Dow-Jones average does not rise on the first day but rises on the
                     second day.
             Q4 : Dow-Jones average does not rise on either day.

Clearly, one of these outcomes must occur, but not more than one can occur at the same time. We can therefore write the sample space as S = {Q1, Q2, Q3, Q4}.

Now, we will consider these two events:

            A : Dow-Jones average rises on the first day.
            B : Dow-Jones average rises on the second day.

Find the intersection, union, and complement of A and B.

Solution: We see that A occurs if either Q1 or Q2 occurs, and, thus,

                 A = {Q1, Q2}           and        B = {Q1, Q3}
The intersection of A and B is the event “Dow-Jones average rises on the first day and rises on the second day.” This is the set of all outcomes belonging to both A and B. Thus,

Finally, the complement of A is the event “Dow-Jones average does not rise on the first day.” This is the set of all basic outcomes in the sample space, S, that do not belong to A. Hence,

Probability and Its Postulates

We consider three definitions of probability
1.    Classical probability
2.    Relative frequency probability
3.    Subjective probability

Classical Probability

Classical probability is the proportion of times that an event will occur, assuming that all outcomes in a sample space are equally to occur. The probability of an event A is

where NA is the number of of outcomes that satisfy the condition of event A  and N is the total number of outcomes in the sample space.

Example: Karlyn Akimoto operates a small computer store. On a particular day she has three Gateway and two Compaq computers in stock. Suppose that Susan Spencer comes into the store to purchase two computers. Susan is not concerned about which brand she purchases—they all have the same operating specifications—so Susan selects the computers purely by chance: Any computer on the shelf is equally likely to be selected. What is the probability that Susan will purchase one Gateway and one Compaq computer?

Solution: Te answer can be obtained using classical probability. To begin, the sample space is defined as all possible pairs of two computers that can be selected from the store. The number of pairs is then counted, as is the number of outcomes that meet the condition—one Gateway and one Compaq. Define the three Gateway computers as, G1, G2, and G3 and two Compaq computers as C1 and C2. The sample space, S, contains the following pairs of computers:

         S = {G1C1, G1C2, G2C1, G2,C2, G3,C1, G3C2, G1G2, G1G3, G2G3, C1C2}

The number of computers in the sample space is 10. If A is the event “One Gateway and one Compaq computer are chosen,” then the number, NA, of outcomes that have one Gateway and one Compaq computer is 6. Therefore, the required probability of event A—one Gateway and one Compaq—is

Formula for Determining the Number of Combinations

The counting process can be generalized by using the following equation to compute the number of combinations of n items taken k at a time:

Thus the number of combinations of the five computers taken two at a time is the number of elements in the sample space:

Example: Suppose that Karlyn’s store now contains 10 Gateway computers, 5 Compaq Computers, and 5 Acer computers. Susan enters the store and wants to purchase 3 computers. The computers are selected purely by chance from the shelf. Now what is the probability 2 Gateway computers and 1 Compaq computer are selected?

 Solution: The classical definition of probability will be used. But in this example the combinations formula will be used to determine the number of outcomes in the sample space and the number of outcomes that satisfy the condition: [2 Gateways and 1 Compaq].

The total number of outcomes in the sample space is

The number of ways that we can select 2 Gateway computers from the 10 available is computed by

Similarly, the number of ways that we can select 1 Compaq computer from the 5 available is computed by

Therefore, the number of outcomes that satisfy event A is

Finally, the probability of A = [2 Gateways and 1 Compaq] is

Relative Frequency Probability

Relative Frequency

We often use relative frequency to determine probabilities for a particular population. The relative frequency probability is the number of events in the population that meet the condition divided by the total number in the population.

Relative Frequency Probability
The relative frequency probability is the limit of the proportion of times that event A occurs in a large number of trials, n:
                    P(A) = 

where nA is the number of A outcomes and n is the total number of trials  or outcomes. The probability is the limit as n becomes large (or approaches infinity).

Example: Probability of Incomes Above $50,000.
Sally Olson is considering an opportunity to establish a new car dealership inDakota Country, which has a population of 15,000 people. Experience from many other dealerships indicates that in similar areas a dealership will be successful if at least 40% of the households have annual incomes over $50,000. She has asked Paul Smith, a marketing consultant, to establish the proportion of family incomes above $50,000, or the probability of such incomes.

Solution: After considering the problem, Paul decides that the probability should be based on the relative frequency. He first examines the most recent census data and finds that there were 54,345 households in Dakota Country and that 31,496 had incomes above $50,000. Paul computed the probability of event A from this source as                

Since Paul knows that there are various errors in census data, he also consulted similar data published by Sales management magazine. From this source he found 55,100 households, with 32,047 having incomes above $50,000. Paul computed the probability of event A from this source as

Since these numbers are close, he could report either. Paul chose to report the probability as 0.58.

Subjective Probability

Subjective probability expresses an individual’s degree of belief about the chance that an event will occur. These subjective probabilities are used in certain management decision procedures.

We can understand the subjective probability concept by using the concept of fair bets. For example, if I assert that the probability of a stock price rising in the next week is 0.5, then I believe that the stock price is just as likely to increase as it is to decrease. In assessing this subjective probability I am not necessarily thinking in terms of repeated experimentation, but instead I am thinking about a stock price over the next week.

Probability Postulates

Let S denote the sample space of a random experiment, Oi, the basic outcomes, and A an event. For each event A of the sample space, S, we assume that P(A) is defined and we have the following probability postulates:

Consequences of the Postulates

We now list and illustrate some immediate consequences of the three postulates.

Example: A charitable organization sells 1,000 lottery tickets. There are 10 major prizes and 100 minor prizes, all of which must be won. The process of choosing winners is such that at the outset each ticket has an equal chance of winning a major prize, and each has an equal chance of winning of wining a minor prize. No ticket can win more than  one prize. What is the probability of winning a major prize with a single ticket? What is the probability of winning a minor prize? What is the probability of winning a minor prize? What is the probability of winning some prize?

Solution: Of the 1,000 tickets, 10 will win major prize, 100 will win minor prizes, and 890 will win no prize. Our single ticket is selected from the 1,000. Let A be the event “Selected ticket wins a major prize,” and let B be the event “Selected ticket wins a minor prize.” The probabilities are

        
The event “Ticket wins some prize” is the union of events A and B. Since only one prize is permitted, these events are mutually exclusive, and

                        

Example: Oil Well Drilling (Probability)
In the early stages of the development of the Hibernia oil site in the Atlantic Ocean, the Petroleum Directorate of Newfoundland estimated the probability to be 0.1 that economically recoverable reserves would exceed 2 billion barrels. The probability for reserves in excess of 1 billion barrels was estimated to be 0.5. Given this information, what is the estimated probability of reserves between 1 and 2 billion barrels?
Solution: Let A be the event “Reserves exceed 2 billion barrels” and B the event “Reserves between 1 and 2 billion barrels.” These are mutually exclusive, and their union, , is the event “Reserves exceed 1 billion barrels.” We therefore have

Probability Rules

We now develop some important rules for computing probabilities for compound events. The development begins by defining A as an event in the sample space, S, with A and its complement, , being mutually exclusive and collectively exclusive.

Complement Rule
Let A be an event andits complement. Then the complement rule is

 Example: Fairselect Inc. is hiring managers to fill four key positions. The candidates are five men and three women. Assuming that every combination of men and women is equally likely to be chosen, what is the probability that at least one women will be selected?

Solution: We will solve this problem by first computing the probability of the complement of A, “No woman is selected,”  and then using the complement rule to compute the probability probabilities of one through three women being selected. Using the method of classical probability,and, therefore, the required probability is

The Addition Rule of Probability

Let A and B be two events. Using the addition rule of probabilities, the probability of their union is

Example: Product Selection (Addition Rule)
A hamburger chain found that 75% of all customers use mustered, 80% use ketchup, and 65% use both. What is the probability that a customer will use at least one of these?

Solution: Let A be the event “Customer uses mustard” and B the event “Customer uses ketchup.” Thus, we have