It is pretty common to say that the chances of getting a head on tossing a fair coin are 50%, the probability of getting a four on rolling a fair dice is 1/6, or the probability that it rains today is 0.1, and so on. These statements make certain claims about the probabilities of the different events. The statements are pretty intuitive and look acceptable, however, what do these statements mean mathematically? What is the meaning of the term probability?
There are multiple different interpretations of probability that explain the meaning of term probability and can also be used to compute the same. Following are a few of them:
The frequency interpretation associates the probability of an event with the relative frequency of occurrence of the event if the experiment is conducted large number of times. For example, if the coin is tossed large number of times, the head will occur in roughly 50% of the cases.
In the subjective, or personal, interpretation of probability, the probability assigned to an event is a belief of the person who assigns the probability. Different person may assign different probabilities to the same event. Because of these reasons, the probability is the subjective probability instead of the true probability of an event.
The classical interpretation associates the probability of an event by the concept of equally likely outcomes. For example, a coin toss can have only two outcomes head or tail, and the both the outcomes are equally likely to occur. Hence, it the probability of occurrence of head is 50%, which is also the case for tail.
Evidently, a particular interpretation cannot be used to explain all kind of observations, moreover, the different interpretations are not even compatible with each other. For example, the frequency interpretation cannot be applied for describing the probability of raining and the classical interpretation is limited by its assumption of equaly likely outcomes. Important point to note is that no single interpretation can explain all the scenarios, and it is also the reason that led to multiple interpretations. Consequently, there is no consensus on a single interpretation either.
The good thing, however, is that the calculus of probability is agnostic to the interpretation of probability. No matter, which interpretation is chosen, the calculus of probability theory remains the same. For example, the Bayes’ theorem does not concern itself with the interpretation of the probability, it does not matter how one interprets the probability, the theorem remains the same.
In summary, there are multiple different interpretations of probability, and no single interpretation can explain all the observations. Fortunately however, the choice of interpretation does not impact the the calculus of probability.
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