There are two basic rules regarding the mathematics of probability:

1. The probability, P, of any event or state of nature occurring is greater than or equal to 0 and less than or equal to 1. That is,

   $$0\le P\left(\text{event}\right)\le 1$$ (2-1)

   A probability of 0 indicates that an event is never expected to occur. A probability of 1 means that an event is always expected to occur.

2. The sum of the simple probabilities for all possible outcomes of an activity must equal 1. Regardless of how probabilities are determined, they must adhere to these two rules.

There are two different ways to determine probability: the objective approach and the subjective approach.

The relative frequency approach is an objective probability assessment. The probability assigned to an event is the relative frequency of that occurrence. In general,

Here is an example. Demand for white latex paint at Diversey Paint and Supply has always been 0, 1, 2, 3, or 4 gallons per day. Over the past 200 working days, the owner notes the frequencies of demand as shown in Table 2.2. If this past distribution is a good indicator of future sales, we can find the probability of each possible outcome occurring in the future by converting the data into percentages.

Thus, the probability that sales are 2 gallons of paint on any given day is $P\left(2\text{ gallons}\right)=$$0.25=25\%.$ The probability of any level of sales must be greater than or equal to 0 and less than or equal to 1. Since 0, 1, 2, 3, and 4 gallons exhaust all possible events or outcomes, the sum of their probability values must equal 1.

Objective probabilities can also be determined using what is called the classical or logical method. Without performing a series of trials, we can often logically determine what the probabilities of various events should be. For example, the probability of tossing a fair coin once and getting a head is

Similarly, the probability of drawing a spade out of a deck of 52 playing cards can be logically set as

When logic and past history are not available or appropriate, probability values can be assessed subjectively. The accuracy of subjective probabilities depends on the experience and judgment of the person making the estimates. A number of probability values cannot be determined unless the subjective approach is used. What is the probability that the price of gasoline will be more than $4 in the next few years? What is the probability that our economy will be in a severe depression in 2020? What is the probability that you will be president of a major corporation within 20 years?

There are several methods for making subjective probability assessments. Opinion polls can be used to help in determining subjective probabilities for possible election returns and potential political candidates. In some cases, experience and judgment must be used in making subjective assessments of probability values. A production manager, for example, might believe that the probability of manufacturing a new product without a single defect is 0.85. In the Delphi method, a panel of experts is assembled to make their predictions of the future. This approach is discussed in Chapter 5.

Events are said to be mutually exclusive if only one of the events can occur on any one trial. They are called collectively exhaustive if the list of outcomes includes every possible outcome. Many common experiences involve events that have both of these properties.

In tossing a coin, the possible outcomes are a head and a tail. Since both of them cannot occur on any one toss, the outcomes head and tail are mutually exclusive. Since obtaining a head and obtaining a tail represent every possible outcome, they are also collectively exhaustive.

Figure 2.1 provides a Venn diagram representation of mutually exclusive events. Let A be the event that a head is tossed, and let B be the event that a tail is tossed. The circles representing these events do not overlap, so the events are mutually exclusive.

The following situation provides an example of events that are not mutually exclusive. You are asked to draw one card from a standard deck of 52 playing cards. The following events are defined:

The probabilities can be assigned to these using the relative frequency approach. There are four 7s in the deck and thirteen hearts in the deck. Thus, we have

These events are not mutually exclusive, as the 7 of hearts is common to both event A and event B. Figure 2.2 provides a Venn diagram representing this situation. Notice that the two circles intersect, and this intersection is whatever is common to both. In this example, the intersection would be the 7 of hearts.

The intersection of two events is the set of all outcomes that are common to both events. The word and is commonly associated with the intersection, as is the symbol $\cap \text{.}$ There are several notations for the intersection of two events:

The notation for the probability would be

The probability of the intersection is sometimes called a joint probability, which implies that both events are occurring at the same time or jointly.

The union of two events is the set of all outcomes that are contained in either of these two events. Thus, any outcome that is in event A is in the union of the two events, and any outcome that is in event B is also in the union of the two events. The word or is commonly associated with the union, as is the symbol $\cup .$ Typical notation for the union of two events would be

The notation for the probability of the union of events would be

In the previous example, the intersection of event A and event B would be

The notation for the probability would be

Also, the union of event A and event B would be

and the probability would be

To see why $P\left(A\text{ or }B\right){=}^{16}{/}_{52}$ and not ${}^{17}{/}_{52}$ (which is $P\left(A\right)+P\left(B\right)$), count all of the cards that are in the union, and you will see there are 16. This will help you understand the general rule for the probability of the union of two events that is presented next.

The general rule for the probability of the union of two events (sometimes called the additive rule) is the following:

To illustrate this with the example we have been using, to find the probability of the union of the two events (a 7 or a heart is drawn), we have

One of the most important probability concepts in decision making is the concept of a conditional probability. A conditional probability is the probability of an event occurring given that another event has already happened. The probability of event A given that event B has occurred is written as $P\left(A|B\right).$ When businesses make decisions, they often use market research of some type to help determine the likelihood of success. Given a good result from the market research, the probability of success would increase.

The probability that A will occur given that event B has occurred can be found by dividing the probability of the intersection of the two events (A and B) by the probability of the event that has occurred (B):

From this, the formula for the probability of the intersection of two events can be easily derived and written as

In the card example, what is the probability that a 7 is drawn (event A) given that we know that the card drawn is a heart (event B)? With what we already know and given the formula for conditional probability, we have

With this card example, it might be possible to determine this probability without using the formula. Given that a heart was drawn and there are 13 hearts with only one of these being a 7, we can determine that the probability is ${}^1{/}_{13}$. In business, however, we sometimes do not have this complete information, and the formula is absolutely essential.

Two events are said to be independent if the occurrence of one has no impact on the occurrence of the other. Otherwise, the events are dependent.

For example, suppose a card is drawn from a deck of cards and it is then returned to the deck and a second drawing occurs. The probability of drawing a seven on the second draw is ${}^4{/}_{52}$ regardless of what was drawn on the first draw because the deck is exactly the same as it was on the first draw. Now contrast this with a similar situation with two draws from a deck of cards, but the first card is not returned to the deck. Now there are only 51 cards left in the deck, and there are either three or four 7s in the deck, depending on what the first card drawn happens to be.

A more precise definition of statistical independence would be the following: Event A and event B are independent if

Independence is a very important condition in probability, as many calculations are simplified. One of these is the formula for the intersection of two events. If A and B are independent, then the probability of the intersection is

Suppose a fair coin is tossed twice. The events are defined as

These events are independent because the probability of a head on the second toss will be the same regardless of the result on the first toss. Because it is a fair coin, we know there are two equally likely outcomes on each toss (head and tail), so

and

Because A and B are independent,

Thus, there is a 0.25 probability that two tosses of a coin will result in two heads.

If events are not independent, then finding probabilities may be a bit more difficult. However, the results may be very valuable to a decision maker. A market research study about opening a new store in a particular location may have a positive outcome, and this would cause a revision of our probability assessment that the new store would be successful. The next section provides a means of revising probabilities based on new information.