4 assignments in Business Statistics due dates listed within description.

Basic Probability

Approaches to Probability

In the previous chapters, we developed charts and graphs to describe the relationships between the characters to find the other data for which we were searching.  The summarizing and displaying of this data is called descriptive statistics.  In Module 5, we now move into a different facet called inferential statistics.

The empirical concept assigns a probability to an event based on information regarding the frequency with which that event occurred.

For example:

We observe a chess championship played between two grand champions.  The two people will play 27 games to decide a champion.  To this point, 20 contests have taken place and player A has won 12 of them.  There are a total number of observations of 20 to this point.  If we want to base a prediction of who will win game #21 solely on what has occurred in the past 20 games (i.e., using relative frequencies of each competitor's success), then we will assign the following probabilities to the players:

Player A has won 12 of 20 games so far, so we predict that there is a 12/20 = 60% chance that player A will win game #21.

Player B has won 8 of 20 games so far, so we predict that there is an 8/20 = 40% chance that player B will win game #21.

The difference between classical probability and the relative frequency approach can be illustrated as follows using Classical probability:

There is a 0.50 or 50% chance that the tossing of a coin will result in a head facing up.  There are two sides to a coin, and if we select heads, there is one favorable outcome.  Thus, the probability is number of favorable outcomes divided by the total number of possible outcomes = 1/2 = 0.50.

Please study the examples in this  Probability Website  for more practice in how to compute probabilities.

Calculating Joint and Conditional Probability

A joint probability measures the probability that two events will occur at the same time.  Consider the following table, which illustrates both the number of students in a given business school major and the gender of students.

Figure 1

What is the probability that a student chosen at random will be a female or a finance major?

From the table, we can determine that there are 200 students in total and 100 of them are female and 45 are finance majors.  Of those finance majors, 25 have already been counted as being female, so our total is: (100 + 45 -25=120).  Hence, the probability is 120/200 = 0.60 or 60%.

Now, look at the same problem from the point of view of probabilities.

Figure 2

The probability of a student chosen at random being female is 0.50 because there are 100 female students out of a total of 200 students.

What is the probability that a student chosen at random will be a female or finance major?

· probability of being female = 0.50

· probability of being a finance major = 0.225

· probability of being both = 0.125

Thus, our result is: 0.50 + 0.225 - 0.125 = 0.60

We can illustrate conditional probability using the same data.  Conditional probability is the probability of an event occurring, given that another event has already occurred.

Consider this example: There are 15 marbles in a jar.  Suppose 9 of them are red and 6 of them are white.  If two marbles are selected at random from the jar, one after the other, without replacement, what is the probability that both are white?

The event that the first one is selected white is W1.  The probability is P(W1)=6/15.  The event that the second marble selected is white is W2.  The probability that the second marble is white, given that the first marble is also white, is P(W2|W1)=5/14.  To determine the probability of two white marbles being selected is P(W1 and W2) = P(W1)P(W2|W1) = (6/15)(5/14) = 0.14.

Please watch the  Joint Probability Table Video  for an example on how you can use Excel to determine a joint probability.

Please watch the  How to Calculate Conditional Probability Video  for a helpful tutorial on how to calculate conditional probability.

Examples of Permutations and Combination

A permutation is the number of ways in which r objects can be taken from a total of n objects.  When the first object is chosen, the second can be chosen from (n-1) objects, and so on, with the result that a general formula can be developed:

Figure 3

For example, candidates for three different political offices are to be chosen from a list of 10 people.  In how many ways can this be done?

Through the logic of permutations, there are 10 choices for the first office.  Then, to go with each of these choices, there are nine possible choices of candidates left for the second office, and to go with both of these choices, there are eight possible candidate choices for the third office.  Thus, there are 10 x 9 x 8 = 720 ways in which three candidates can be chosen from a set of 10.

The formula uses factorials to determine this:

Figure 4

A combination reflects the same principle as a permutation, but in this case, the actual order of items taken does not matter.  The number of combinations is less than the number of permutations because the order of selection doesn't matter.

For example, we have three letters, XYZ, and we wish to take two letters from this set of three.

How many permutations are there?  The formula tells us that there are 3!/(3-2)! = 6.  Since this is a small number, we can write them all out:

XY, YX, XZ, ZX, YZ, ZY

How many combinations of two letters can we draw from these three letters?  Since the order of the pair does not matter, there are only three combinations: XY, XZ, and YZ, because XY would be the same pair as YX.

The number of combinations of n elements taken r at a time is described by the general formula:

Figure 5

For our previous political office example,

Figure 6

Visit the  Combinations and Permutations Website  for some helpful real-life examples of how to determine combinations and permutations.

Relative Frequency Approach

The second type of object probability is relative frequency, also called empirical probability.  The small difference is the comparison on the number of times that the event occurs in proportion to the known number of trials.  The reliability of the empirical probability approach is based on the law of large numbers.  The critical concept is that the empirical probability becomes equivalent to its true probability when the number of trials is very large.

We toss a coin 1000 times and get 550 heads and 450 tails.  We determine probabilities based on the number of times the event has occurred in the past divided by the total number of observations.

For this example, to obtain the probability of a head on the next toss: 550/1000 = 0.55 or 55%; and a tail: 450/1000 = 0.45, or 45%.

When you do not have the advantage of a large number of trials, with little experience or no information upon which to calculate probability, one has to option to approach it from a subjective viewpoint.  In this case, we use the terms subjective probability.  Examples of subjective probability are in the news all the time.  Examples of this, similar to those in the textbook, would be the odds that your favorite team will be playing in the championship game this year or next year.

Additional Rules for Computing Probabilities

Rules of Addition

We will examine the general rule of addition, special rule of addition, and the complement rule.  The reason for having the rule of addition is to cover the instances in which the outcomes of any experiment are not mutually exclusive.  Using a retail store example, the percentage of people who bought a loaf of bread is not mutually exclusive from those who bought a gallon of milk.  Adding the individual probabilities together would double count those who bought both a loaf of bread and a gallon of milk.  The joint probability of both events happening is different than the probability that either would happen.  Pages 154 through 157 cover other examples.

Rules of Multiplication

As in the rules of addition, we have rules of multiplication.  The general rule of multiplication applies when two events are not independent and, therefore, dependent.  This introduces the concept of conditional probability.  Conditional probability estimates the probability that a particular secondary event occurs, given that a specified primary event has already occurred.  This general rule is used to find the joint probability between those two events.

The formula looks like this: P(A and B) = P(A)P(B|A)

The special rule for multiplication applies when the two events are independent.  A very simple example clearly illustrates the principle.  The textbook uses the coin flip of two different coins and the resulting outcomes to illustrate this principle.

The formula for this special rule is P (A and B) = P(A)P(B).

Contingency Tables

Contingency tables are used to categorize data sets by setting up a matrix of the outcomes of two or more identical identifiable characteristics.  In these particular cases, we now begin to combine the rules of addition and multiplication.

One of the tools used to help illustrate these conditions are tree diagrams.  The process is broken down into stages and each layer of branches represents the next stage of the process.  Each of the branches then becomes weighted by their respective probabilities.  The textbook provides a very good example of what tree diagrams look like on page 165.

Visit the  Practice with Tree Diagrams Website  for some additional practice problems that you can work through to understand the concept of tree diagrams.

The previous relationships between events were independent and dependent.  We will now take a look at what happens when the outcomes are mutually exclusive.  Bayes theorem introduces the concepts of prior probability and posterior probability.  Prior probability being the initial probability deduced from current evidence available.  Posterior probability then becomes a recalculation based on new information.  The application of Bayes theorem, using the example the textbook, applies when deciding between vendors based on reliability of their products that they have produced for you.  This is an example of the many ways that businesses can utilize calculations derived in the 18th century to solve 21st-century business problems.

For a fun review of the concepts we have learned about in this lesson, please check out the probability song by watching the  Probability Video .