The posterior probability is that which is calculated based on data already known after a process or experiment.
The a posteriori probability is, then, the one that is not estimated based on conjecture or some prior knowledge regarding the distribution of a probability, as in the prior probability.
To understand it better, let's look at an example.
Suppose a company is developing a new toiletries product, for example a shampoo.Thus, the company evaluates a group of volunteers to see if any percentage of them develop dandruff after using the product.
Thus, it is obtained, for example, that the posterior probability that an adult man will develop dandruff when trying this new product is 2%.
Instead, an example of a priori probability occurs when, before rolling a die, we assume that there is the same probability that any of the six numbers will roll as a result, that is, 1/6.
History of probability
Posterior probability and Bayes theorem
To solve exercises with posterior probabilities, Bayes' theorem is usually used, whose formula is the following:
In the formula above, B is the event we have information about and A (n) are the various conditional events. That is, in the numerator we have the conditional probability, which is the possibility of an event B occurring given that another event An has taken place. While in the denominator we observe the sum of the conditioned events, which would equal the probability total occurrence of event B, assuming that none of the possible conditioned events is neglected.
Better let's see, in the next section, an example so that it is better understood.
Example of a posteriori probability
Suppose we have 4 classrooms that have been evaluated with the same exam.
In the first group or classroom, which we called A, 60% of the students passed the assessment, while in the rest of the classrooms, which we will call B, C and D, the percentage of passing was 50%, 56% and 64%, respectively. These would be posterior probabilities.
Another fact to take into account is that classrooms A and B have 30 students, while classrooms C and D have 25 each. So, if we choose, among the exams of the four groups, a random evaluation and it turns out to have a passing grade, what is the probability that it belongs to classroom A?
For its calculation, we will apply Bayes' theorem, An being the conditional event that the exam belongs to a student in classroom A and B the fact that the grade is passing:
P [An / B] = (0.6 * 30/110) / (* (30/110) + * (30/110) + * (25/110) + * (25/110))
P [An / B] = 0.1636 / 0.5727 = 0.2857
It should be noted that we divide the number of students in classroom X by the total number of students in the four groups to find out the probability that the student is in classroom X.
The result tells us that there is a probability of 28.57%, approximately, that, if we choose an exam at random and it has a passing grade, it will be from classroom A.