2.3. Three conceptual approaches to probability

There are three conceptual approaches to probability:

1. Classical probability

2. Relative frequency concept of probability

3. Subjective probability.

2.3.1. Classical probability

Classical probability assumes that all outcomes in the sample space are equally likely to occur. It was developed originally in the analysis of gambling problems, where the assumption of equally likely outcomes often is reasonable. When the assumption of equally likely outcomes is used a basis for assigning probabilities, the approaches is referred to as the classical method.

According to the classical method, the probability of a single event is equal to one divided by the total number of outcomes for experiment. On the other hand, the probability of a compound event A is equal to number of outcomes favourable to event A, divided by the total number of outcomes for the experiment.

2.3.2. Relative frequency concept of probability

The difference between classical probability and relative frequency probability is that classical method assumes that certain outcomes are equally (such as the outcomes when a die is rolled) while relative frequency method relies on actual experience to determine the likelihood of outcomes. In relative frequency method, one might actually roll a given die 1000 times and observe the relative frequencies and use these frequencies to determine the probability of an outcome. This method of assigning a probability to an event is called the relative frequency concept of probability.

Definition:

If an experiment is repeated n times and an event A is observed f times, then, according to the relative frequency concept of probability:

Because relative frequencies are determined by performing an experiment, the probabilities calculated using relative frequencies may change almost each time an experiment is repeated. But the variation in probabilities will be small if the sample size is large.

2.3.3. Subjective probability

Subjective probability uses a probability value based on an educated guess or estimate, employing opinions.

In subjective probability, a person or group makes an educated guess at the chance that an event will occur. This guess is based on person’s experience and evaluation of solution. For example, a sportswriter may say that there is a 65% probability that the Milan will win championship cup next year. A doctor might say that on the basis of his diagnosis, there is a 40% chance the patient will need an operation. A seismologist might say there is a 60% probability that an earthquake will occur in a certain area.

Definition:

Subjective probability is the probability assigned to an event based on subjective judgment, experience, information, and belief.

All three types of probability (classical, relative frequency, and subjective) are used to solve variety of problems in business, economics, engineering, and other fields.

2.4. Probability and its postulates

Probability, which gives the likelihood of occurrence of an event, is denoted by P.

Let S denote the sample space of a random experiment, the basic outcomes, and A an event. The probability that an event A will occur is denoted by .

The probability has the following important properties:

1. If A is any event in the sample space S, then .

An event that can not occur has zero probability; such event is called an impossible event. An event that is certain to occur has a probability equal to 1 and is called sure event.

For impossible event M: P (M) =0

For a sure event C: P (C) =1

2. Let A be an event in sample space S, and let denote the basic outcomes. Then

,

where the notation implies that the summation extends over all the basic outcomes in A.

3. The sum of the probabilities for all the basic outcomes in the sample space always is 1.Thus

.