What is Probability?

Most of the decisions made by individuals and businesses happen in environments where random events make the outcome uncertain. Let's consider a simple example where your company is about to market a new product and needs to decide what price to charge. There are already some close substitute products in the market, so you need to understand how your competitors will react to the introduction of the new product. You also need to take into account how consumers will respond to different price points. It would be great for your company if you could determine exactly how the competition and consumers would behave, but that is probably not possible because you most likely do not have that information. This limited knowledge about the outcome is what makes the decisions of consumers and competitors random from your company's perspective. The company must find a way to assess the likelihood of each of these random events. You can help do that by assigning probabilities to them. These probabilities can be calculated from data the company has collected on past decisions of competitors and consumers, or from subjective beliefs about their behavior.

Set Theory

Before we can define probability, study its most important properties, and start applying them to real world problems, we need to review a little bit of set theory. The reason is simple: probability is defined in terms of and calculated for sets of possible realizations of a random experiment. A set can be thought of as a well-specified collection of things, called its elements. A set can have a finite or infinite number of elements. In business statistics, you want a set to include as many possible outcomes as you can in a set so you can make a well-informed plan for the success of your new venture. 

The mathematical definition of a set requires us to identify its elements. This can be done by listing all of its elements, when that is feasible, or by describing their properties. For instance, when a company can produce a maximum of 5 hand-crafted entertainment stands per month, the set of all possible monthly output levels is { 0 , 1 , 2 , 3 , 4 , 5 } .

If company A wants to know the average income of its potential customers, it would need to have access to the set of data { x : x   is the income of a consumer who can potentially purchase the product of company A } and then take the average of the elements of this set.

The notation we are using is standard in set theory, and can be written in general as { x : x   satisfies a certain property } where the colon means “such that.” This reads “A is the set of elements x such that x satisfies a certain property.” Here are a few more examples:

  1. R + = { x : x   is a real number and   x 0 } is the set of all real numbers greater than or equal to zero. This set can be used to represent all the prices that a company could possibly charge.

  2. D = is the empty set. By definition, it has no elements.

  3. A consumer who has $100 to spend on two items whose prices are $2 and $3 can only purchase quantities x 1 and x 2 of those two items that belong to the set B = { ( x 1 x 2 ) : 2 x 1   + 3 x 2   100 } . This set consists of all pairs of quantities ( x 1 , x 2 ) such that total expenditure 2 x 1 + 3 x 2 is less than or equal to income $100.

We use the symbol to designate the property of being an element of a set. For instance, the consumer in example 3 above can afford to purchase 20 units of each item. In that case, her expenditure is equal to her income: 2 × 20 + 3 × 20 = 100 .

Therefore, we write ( 20 , 20 ) B . She cannot afford to purchase 30 units of each item, though, so ( 30 , 30 ) B .

If A and B are two arbitrary sets and every element of A is also an element of B, then A is a subset of B, written A B . For instance, if A = { 1 , 2 , 3 , 4 } and B = { 1 , 2 , 3 , 4 , 5 } , then A B . On the other hand, A is not a subset of B, written A B , if there exists at least one element of A that doesn’t belong to B. In the example where a consumer purchases two items, if we denote the consumer’s income by m and the prices of items 1 and 2 by p 1 and p 2, respectively, then { ( x 1 , x 2 ) : p 1 x 1 + p 2 x 2 m } { ( x 1 , x 2 ) : p 1 x 1 + p 2 x 2 > m } .

A useful way to visualize relationships between sets is through Venn diagrams, where sets are represented as geometrical figures. In the picture below, circle A is completely inside circle B, which represents the situation where A B .

Basic Set Operations

We are now ready to state some basic set operations that we will need in our study of probabilities. Given two sets A and B, we define the following operations: union, intersection, and complement.

  1. The union of two sets A and B is the set of those elements (and only those elements) that belong to A, or to B or both. In set notation: A B { x : x A   o r   x B } . In the Venn diagram below, the union of A and B is the entire blue area.

    Union of A and B
  2. The intersection of two sets A and B is the set of those elements (and only those elements) that belong to both A and B. In set notation: A B { x : x A   a n d   x B } . Two sets are disjoint or mutually exclusive if their intersection is the empty set. In the Venn diagram below, the intersection of A and B is the darker blue area, so A and B are not disjoint.

    Intersection of A and B
  3. The complement of a set A, denoted A C , is the set of all the elements that do not belong to A. In set notation: A C { x : x A } . In the Venn diagram below, the complement of A is the gray area.

    The complement of A is the gray area
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