Bayes’ theorem

Bayes’ theorem is one of the pillars of probability. This is a theory put forward by Thomas Bayes (1702-1761) in the 18th century. But what exactly is the scientist trying to explain? According to the Royal Spanish Academy, the probability It expresses, in a random process, the ratio between the number of favorable cases and the number of possible cases.

So, Many theories have been developed around probability that govern us today. When we go to the doctor, he prescribes what is most likely to be good for us, advertisers dedicate their campaigns to people who are most likely to purchase the product they want to promote, we choose the route that is most likely to take us the least amount of time.

One of the most famous laws of probability is law of total probability . To begin, we must take into account what the law of total probability is about. To understand it, let’s take an example.

Let’s say 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 person chosen at random from the active population in this country is unemployed?

According to probability theory, the data would be expressed as follows:

The probability that the person was women: P.M)The probability that the person was man: P(H)

Knowing that 39% of the population are women, then we deduce that: P(M) = 0.39.

Then, we understand that: P(H) = 1 – 0.39 = 0.61. The silver problem also gives us the conditional probabilities:

Probability that a person is unemployed, knowing that he is a woman —> P (P|M) = 0.22Probability that a person is unemployed, knowing that he is a man —- P (P|H) = 0.14

In this way, using the law of total probability we will have:

P (P) = P(M)P(P|M) + P(H)P(P|H)

P(P) = 0.22 × 0.39 + 0.14 × 0.61

P(P) = 0.17

So, The probability that a person chosen at random is unemployed will be 0.17. We observe that the result is between the two conditional probabilities (0.22<0.17<0.14). Furthermore, it is closer to men because they are the majority in the population of this invented country.

Bayes’ theorem

Well, now suppose that an adult is chosen at random to fill out a form and it is observed that he or she does not have a job. In this case, and taking into account the previous example, what is the probability that this randomly chosen person is a woman -P (M|P)-?

To solve this problem we will apply the Bayes theorem. Thus, this theorem is used to calculate the probability of an event given advance information about that event. We can calculate the probability of an event A, also knowing that this event A meets a certain characteristic (B) that conditions its probability.

In this case, we are talking about the probability that the person randomly chosen to fill out a form is a woman. But also, This probability will not be independent of whether the selected person is unemployed or not..

The formula of Bayes’ theorem

Like any other theorem, to calculate probability we need a formula. In these types of events, The formula is defined like this:

It seems complicated but everything has an explanation. Let’s go by parts. What does each letter mean?

To start, b is he event about which we have prior informationFor its part, the letter A(n) refers to the different conditioned events.In the part of the numerator we have the conditional probability. This refers to the probability that something (event A) occurs, knowing that another event (B) also occurs. is defined as P(A|B) and is expressed as: The probability of A given B. In the denominator, we have the equivalent of P(B). See the previous section.

An example:

So, returning to the previous example, suppose that an adult is chosen at random to fill out a questionnaire and it is observed that he does not have a job (He is unemployed). What will be the probability that this chosen person is a woman?

Well, taking into account the previous example, we know that 39% of the active population are women. We know, then, that the rest are men. Furthermore, we know that the percentage of unemployed women is 22% and of men it is 14%.

Finally, we also know that The probability that a person chosen at random is unemployed is 0.17.. So, if we apply the formula of Bayes’ theorem, the result we will obtain is that there is a 0.5 probability that a person chosen at random, from among all those who are unemployed, is a woman.

P(M|P) = (P(M) * P (P|M)/P(P)) = (0.22 * 0.39)/0.17 = 0.5

Let’s say goodbye to this probability article by referring to one of the most frequent confusions regarding probability. This oscillates between 1 and 0, never leaving these margins; where 1 is the probability of a certain event and 0 is the probability of an impossible event..

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