What is conditional probability and how to calculate it correctly?

Often in life we are faced with the fact that we need to assess the chances of an event. Whether it is worth buying a lottery ticket or not, what will be the sex of the third child in the family, whether tomorrow will be clear weather or again rain - there are countless examples of such examples. In the simplest case, the number of favorable outcomes should be divided by the total number of events. If there are 10 winning tickets in the lottery, and only 50 of them, then the chances of getting a prize are 10/50 = 0.2, that is 20 versus 100. And what if there are several events and they are closely related? In this case, we are not interested in the simple but conditional probability. What is this value and how it can be counted - this is exactly what will be told in our article.

The concept of

A conditional probability is the chance of an event occurring, provided that another event associated with it has already occurred. Consider a simple example with a coin toss. If the draw was not yet, then the chances of an eagle or tails falling are the same. But if you fold a coin five times in a row, then you agree to expect the 6th, 7th, and even more so the 10th repetition of such an outcome will be illogical. With every repetition of the eagle's fall, the chances of the appearance of tails grow and sooner or later it will fall out.

The conditional probability formula

Let's now figure out how this value is calculated. We denote the first event by B, and the second by A. If the odds of approach B are different from zero, then the following equality will be valid:

P (A | B) = P (AB) / P (B), where:

• P (A | B) is the conditional probability of total A;
• P (AB) - probability of joint occurrence of events A and B;
• P (B) is the probability of the event B.

By slightly converting this relation, we obtain P (AB) = P (A | B) * P (B). And if you apply the induction method, you can derive the product formula and use it for an arbitrary number of events:

P (A ₁ , A ₂ , A ₃ , ... A _n ) = P (A ₁ | A ₂ ... A _n ) * P (A ₂ | A ₃ ... A _n ) * P (A ₃ | A ₄ ... A _n ) ... P (A _n-1 | A _n ) * P (A _n ).

Practice

To make it easier to understand how the conditional probability of the event is calculated , let us consider a couple of examples. Suppose there is a vase in which there are 8 chocolates and 7 mints. In size, they are the same and at random two of them are successively pulled out. What are the chances that both of them will turn out to be chocolate? We introduce the notation. Let the result of A mean that the first candy is chocolate, the result of B is the second chocolate candy. Then we get the following:

P (A) = P (B) = 8/15,

P (A | B) = P (B | A) = 7/14 = 1/2,

P (AB) = 8/15 x 1/2 = 4/15 ≈ 0.27

Let's consider one more case. Suppose there is a two-child family and we know that at least one child is a girl. What is the conditional probability that these parents do not yet have boys? As in the previous case, we start with the notation. Let P (B) be the probability that there is at least one girl in the family, P (A | B) is the probability that the second child is also a girl, P (AB) are the chances that there are two girls in the family. Now let's make the calculations. In total there can be 4 different combinations of the sex of children, and in only one case (when there are two boys in the family), there will be no girls among the children. Therefore, the probability P (B) = 3/4, and P (AB) = 1/4. Then, following our formula, we get:

P (A | B) = 1/4: 3/4 = 1/3.

Interpret the result as follows: if we were not aware of the field of one of the children, the chances of the two girls would be 25 to 100. But since we know that one child is a girl, the probability that there are no boys in the family grows to one Third.