The Bayes theorem is one of the pillars of probability. This is a theory presented by Thomas Bayes (1702–1761) in the 18th century. But what exactly was this scientist trying to explain?According to the Royal Spanish Academy, the probability expresses, in a random process, the relationship between the number of favorable cases and the number of possible cases.
Many theories about probabilities have been developed, governing us to this day. When we go to the doctor, he prescribes what is most likely to solve our problem, advertisers dedicate their campaigns to the people most likely to buy the product. To promote, we’ve chosen the route that’s likely to take less time.
- One of the most well-known probability laws is the law of total probability.
- To begin with.
- We need to consider what the law of total probability is.
- To understand this.
- Let’s take an example.
Suppose that, in a random country, 39% of the population are women. We also know that 22% of women and 14% of men are unemployed. So what is the probability (P) that a randomly selected workforce person in this country is unemployed?
According to probability theory, the data would be expressed as follows:
Knowing that 39% of the population are women, we induce that: P (M) – 0. 39.
Therefore, we understand that: P (H) – 1-0. 39 – 0. 61 The problem presented also gives us the conditional probabilities:
So, using the total probability law, we’ll have
P (P) – P (H) P (P H) P (H) P (P H)
P (P) – 0. 22 – 0. 39 0. 14 – 0. 61
P (P) – 0. 17
Thus, the probability that a randomly selected person is unemployed is 0. 17 It is observed that the result is between the two conditional probabilities (0. 22 Bayes Theorem
Now suppose that an adult is randomly selected to fill out a form and finds that he has no job, in this case, and in the light of the example above, what is the probability that this randomly selected person is a woman?-P (MP) -?
To solve this problem we will apply Bayes’ theorem, so this theorem is used to calculate the probability of an event by having previous information about that event, we can calculate the probability of an event A, also knowing that event A responds to a certain characteristic (B), which determines its probability.
In this case, we are talking about the probability that the person randomly selected to complete a form is a woman, besides, this probability will not be independent of whether the selected person is unemployed or not.
Like any other theorem, to calculate probability we need a formula, in this type of event, the formula is defined as:
It sounds complicated, but everything has an explanation. Let’s go in parts. What does each letter mean?
So, going back to the example above, let’s say you choose a random adult to complete a questionnaire and you see that you don’t have a job (you’re unemployed) What’s the probability that this person is a woman?
Well, considering the example above, we know that 39% of the workforce is women, so we know that others are men. We also know that the percentage of unemployed women is 22% and 14% of men.
Finally, we also know that the probability of a randomly selected person being unemployed is 0. 17, so if we apply the Bayes theorem formula, the result we get is that there is a 0. 5 chance that a randomly selected person, among all those unemployed, is a woman.
P (M P) – (P (M) – P (P) M) / P (P)) – (0. 22 – 0. 39) / 0. 17 – 0. 5
We say goodbye to this article on probabilities referring to one of the most common confusions about probability, 1 to 0, without ever going out of these margins, with 1 being the probability of a safe event and 0 being the probability of an event.