Two basic types of probability can be defined: objective probability and subjective probability . First, we will consider what constitutes objective probability.
Consider a referee's flipping a coin before a football game to determine which team will kick off and which team will receive. Before the referee tosses the coin, both team captains know that they have a .50 (or 50%) chance (or probability) of winning the toss. None of the onlookers in the stands or anywhere else would argue that the probability of a head or a tail was not .50. In this example, the probability of .50 that either a head or a tail will occur when a coin is tossed is called an objective probability . More specifically , it is referred to as a classical or a priori (prior to the occurrence), probability , one of the two types of objective probabilities.
Objective probabilities that can be stated prior to the occurrence of an event are classical , or a priori, probabilities .
A classical, or a priori, probability can be defined as follows : Given a set of outcomes for an activity (such as a head or a tail when a coin is tossed), the probability of a specific (desired) outcome (such as a head) is the ratio of the number of specific outcomes to the total number of outcomes . For example, in our coin- tossing example, the probability of a head is the ratio of the number of specific outcomes (a head) to the total number of outcomes (a head and a tail), or 1/2. Similarly, the probability of drawing an ace from a deck of 52 cards would be found by dividing 4 (the number of aces) by 52 (the total number of cards in a deck) to get 1/13. If we spin a roulette wheel with 50 red numbers and 50 black numbers , the probability of the wheel landing on a red number is 50 divided by 100, or 1/2.
These examples are referred to as a priori probabilities because we can state the probabilities prior to the actual occurrence of the activity (i.e., ahead of time). This is because we know (or assume we know) the number of specific outcomes and total outcomes prior to the occurrence of the activity. For example, we know that a deck of cards consists of 4 aces and 52 total cards before we draw a card from the deck and that a coin contains one head and one tail before we toss it. These probabilities are also known as classical probabilities because some of the earliest references in history to probabilities were related to games of chance, to which (as the preceding examples show) these probabilities readily apply.
The second type of objective probability is referred to as relative frequency probability . This type of objective probability indicates the relative frequency with which a specific outcome has been observed to occur in the long run. It is based on the observation of past occurrences. For example, suppose that over the past 4 years , 3,000 business students have taken the introductory management science course at State University, and 300 of them have made an A in the course. The relative frequency probability of making an A in management science would be 300/3,000 or .10. Whereas in the case of a classical probability we indicate a probability before an activity (such as tossing a coin) takes place, in the case of a relative frequency we determine the probability after observing, for example, what 3,000 students have done in the past.
Objective probabilities that are stated after the outcomes of an event have been observed are relative frequency probabilities .
The relative frequency definition of probability is more general and widely accepted than the classical definition. Actually, the relative frequency definition can encompass the classical case. For example, if we flip a coin many times, in the long run the relative frequency of a head's occurring will be .50. If, however, you tossed a coin 10 times, it is conceivable that you would get 10 consecutive heads. Thus, the relative frequency (probability) of a head would be 1.0. This illustrates one of the key characteristics of a relative frequency probability: The relative frequency probability becomes more accurate as the total number of observations of the activity increases . If a coin were tossed about 350 times, the relative frequency would approach 1/2 ( assuming a fair coin).
Relative frequency is the more widely used definition of objective probability .
When relative frequencies are not available, a probability is often determined anyway. In these cases a person must rely on personal belief, experience, and knowledge of the situation to develop a probability estimate. A probability estimate that is not based on prior or past evidence is a subjective probability . For example, when a meteorologist forecasts a "60% chance of rain tomorrow," the .60 probability is usually based on the meteorologist's experience and expert analysis of the weather conditions. In other words, the meteorologist is not saying that these exact weather conditions have occurred 1,000 times in the past and on 600 occasions it has rained, thus there is a 60% probability of rain. Likewise, when a sportswriter says that a football team has an 80% chance of winning, it is usually not because the team has won 8 of its 10 previous games. The prediction is judgmental, based on the sportswriter's knowledge of the teams involved, the playing conditions, and so forth. If the sportswriter had based the probability estimate solely on the team's relative frequency of winning, then it would have been an objective probability. However, once the relative frequency probability becomes colored by personal belief, it is subjective.
Subjective probability is an estimate based on personal belief, experience, or knowledge of a situation .
Subjective probability estimates are frequently used in making business decisions. For example, suppose the manager of Beaver Creek Pottery Company (referred to in chapters 2 and 3) is thinking about producing plates in addition to the bowls and mugs it already produces. In making this decision, the manager will determine the chances of the new product's being successful and returning a profit. Although the manager can use personal knowledge of the market and judgment to determine a probability of success , direct relative frequency evidence is not generally available. The manager cannot observe the frequency with which the introduction of this new product was successful in the past. Thus, the manager must make a subjective probability estimate.
This type of subjective probability analysis is common in the business world. Decision makers frequently must consider their chances for success or failure, the probability of achieving a certain market share or profit, the probability of a level of demand, and the like without the benefit of relative frequency probabilities based on past observations. Although there may not be a consensus as to the accuracy of a subjective estimate, as there is with an objective probability (e.g., everyone is sure there is a .50 probability of getting a head when a coin is tossed), subjective probability analysis is often the only means available for making probabilistic estimates, and it is frequently used.
A brief note of caution must be made regarding the use of subjective probabilities. Different people will often arrive at different subjective probabilities, whereas everyone should arrive at the same objective probability, given the same numbers and correct calculations. Therefore, when a probabilistic analysis is to be made of a situation, the use of objective probability will provide more consistent results. In the material on probability in the remainder of this chapter, we will use objective probabilities unless the text indicates otherwise .