Theoretical and Experimental Probability

A probability is a numerical measure of the chances of an event occurring. Mostly the events are favorable outcomes like getting a head or tail in a toss, or getting a favorite number to be rolled from a die, winning a game etc. But sometimes the events may not be ‘favorable’ outcomes like number of persons likely to be affected by an epidemic, number of trees likely to fall, number of death likely in an accident etc. Thus a probability is an important guidance in many situations. The probabilities of different events can be determined in two ways. One way is by working out with the help of mathematical formulas and the second by conducting actual experiments. The probability determined by the first method is called ‘theoretical probability' and the second method is known as ‘experimental probability’.

Theoretical Probability and Experimental Probability

We have given a brief introduction about theoretical probability and experimental probability. In either case, the basic formula for determining a probability is, P(x) = number of favorable outcomes for the event ‘x’ to occur/ total number of all possible outcomes.In the theoretical probabilities, the events are classified, the outcomes are mathematically defined. For example, you roll two fair dice and want to determine the probability of getting a ‘sum of 8’. The basics are worked out like ‘sum of ‘8’ is possible with two dice in different ways like 2 + 6, 3 + 5, 4 + 4, 5 + 3, 6 + 2. Now the probability of getting all the first and second numbers are worked out and finally transformed to the required probability, again by mathematical methods. Hence, at each and every stage ‘theory’ is employed and thus this method of determining the probability is named as ‘theoretical probability’.

On the other hand, for the same probability, the practical method just requires throwing a fair die and noting down the respective numbers and finally listing out the trials in which the sum was 8. Here the method is more physical and manual. Since it is a practical experiment, a probability determined this way is called as an experimental probability.

Theoretical Probability

As explained earlier a theoretical probability is determined by mathematical methods using theories and formulas. Of course, there are some exceptions which we will deal at a later stage.

To determine the theoretical probability, firstly the case must be studied thoroughly to identify the methods to be used. The study includes the type of events, the distribution of probability in different cases, the best method of approach etc. In certain cases where conditional probabilities are involved (that is probability of a ‘probable’ event, the probability of which is known), a known probability is needed to determine the required probability. The ‘known’ probability may be a theoretical one (like rolling a die for the number 4) or may be an experimental probability arrived by rigorous experiments (like what section of women aged above 45 is prone to breast cancer). Thus, a theoretical probability is a result of mathematical work done in the right way.

Experimental Probability

As the name suggests the probability determined by actually conducting the trials of the required events. A theoretical probability is the ratio of the number of successful trials in an actual experiment to the total number of trials made. This is done even for cases of simple probability distributions like tossing a coin. But it is still done to conclude how far the results of experimental probabilities agree with the results based on theoretical probabilities. It may be interesting to note that in most cases the two results are fairly close when the number of trials is large. The accuracy of experimental probability becomes more with increase in number of trials.

However, an experimental probability must be conducted in a fair and unbiased manner. For example, in case of tossing a coin or a die we assume that the coin or the die to be fair. In case of theoretical probability it is just an assumption and we proceed. In practice, these aspects are to be confirmed practically. The object of the experiment is to get unbiased results which only can give the true results. Let us take a case study for example. A team makes a survey to determine the probability of a person being infected with a certain virus. The team visits a number of hospitals in the city from morning to evening for taking the survey and makes a report. Clearly the report is biased. It is obvious most of the infected persons go to the hospitals for treatment. Hence the data mostly includes the section of people who are affected. The section of the people who are unaffected is not at all represented. The team should have visited different localities apart from hospitals to collect the data. There may be many persons who  are not affected or mildly affected who do not need a hospital treatment. Further, the team makes the survey at the convenient time suited to them. There is a good chance that many patients might visit the hospital in the evenings and nights and that population is not taken into account. Therefore, the report will be far from factual.

There could be some practical difficulties in conducting an experiment. For example, to determine the probability of a person is employed or unemployed it will be difficult to get a suitable sample size. In such cases, you have to very carefully design a stimulating method. Even in such cases, the accuracy may not be good, unless a big population from all sections is considered.

Similarly some experimental probabilities are determined based on past records. As an example to judge the probability of the winner in a horse race, the history of all the participating horses and jockeys are to be studied thoroughly along with their past performances in different situations.

Theoretical vs Experimental Probability

Let us make a  table of comparison between theoretical probability and experimental probability.

   Theoretical Probability                                                                                   
                                  Experimental Probability                                       
 Probability is determined by mathematical methods.  Probability is determined by observing the successful events in an actual experimental
 Binding factors are only assumed.  Binding factors are to be enforced in actual practice.
Probability of any event can be predicted by correctly identifying the probability distribution and adopting appropriate mathematical methods.  Probability of certain events cannot be determined by actual experiments. Some probabilities can only be estimated by past records.
 Number of trials (no of variables) can be small for normalizing the probability.  Requires a large number of trials to get an accuracy close to theoretical probability
 Process cost is practically nil.  Substantial cost is needed for actual experiments.