what is theoretical probability?

Theoretical probability is the probability something will happen based on a calculation. The calculation compares the number of ways that thing will happen to the total number of possible outcomes. 
For example, a regular die has six faces with the numbers 1 through 6 on them. Since only one of the six faces has a 4, the theoretical probability of rolling a 4 is 1/6. Similarly, the theoretical probability of rolling a number smaller than 6 is 5/6, because five of the six faces have numbers smaller than 6.
Theoretical probability is different from experimental probability. In experimental probability, first someone would do an experiment like roll a die 100 times and record the results. Then they would calculate the experimental probability of rolling a 4 based on the number of times a 4 was actually rolled in the experiment. Because results of experiments will likely be different each time an experiment is done, the experimental probabilities will also be different. They might sometimes be the same as the theoretical probability, and they might sometimes be different. The theoretical probability of rolling a 4, however, will always be 1/6.


There are various types of probability. Experimental (empirical) probability involves performing a number of trials and recording the outcomes. Subjective probability involves pattern recognition and professional judgement.
Theoretical probability is a type of probability that does not require performing an experiment. The probabilities of various events can be calculated just from the knowledge of the system in question. The theoretical probability is the number of outcomes in the event space (the number of outcomes that have the required attribute) divided by the number of outcomes in the sample space (the total number of possible outcomes.)
Examples include the probability of getting a certain number of heads from n flips, the probability of selecting a red ball then a blue ball from an urn, the probability of being dealt a straight flush in draw poker, etc...
Compare to empirical probability where you would use methods like a Monte Carlo simulation to compare many different trials of an experiment.
We can use theoretical probability to check whether a coin or die is "fair" by comparing the theoretical probabilities to the recorded empirical probabilities.
http://mathworld.wolfram.com/Probability.html

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