Often in life we are faced with the need to assess the chances of an event occurring. Whether it is worth buying a lottery ticket or not, what will be the gender of the third child in the family, whether tomorrow the weather will be clear or it will rain again - there are countless such examples. In the simplest case, you should divide the number of favorable outcomes by the total number of events. If there are 10 winning tickets in the lottery, and there are 50 in total, then the chances of getting a prize are 10/50=0.2, that is, 20 against 100. But what if there are several events, and they are closely related? In this case, we will no longer be interested in simple, but in conditional probability. What this value is and how it can be calculated - this will be discussed in our article.
Concept
Conditional probability is the chance of a particular event occurring, given that another related event has already happened. Consider a simple example withtossing a coin. If there has not been a draw yet, then the chances of getting heads or tails will be the same. But if five times in a row the coin lay with the coat of arms up, then agree to expect the 6th, 7th, and even more so the 10th repetition of such an outcome would be illogical. With each repeated heading, the chances of tails appearing grow and sooner or later it will fall out.
Conditional probability formula
Let's now figure out how this value is calculated. Let us denote the first event as B, and the second as A. If the chances of occurrence of B are different from zero, then the following equality will be valid:
P (A|B)=P (AB) / P (B), where:
- P (A|B) – conditional probability of outcome A;
- P (AB) - the probability of joint occurrence of events A and B;
- P (B) – probability of event B.
Slightly transforming this ratio, we get P (AB)=P (A|B)P (B). And if we apply the method of induction, then we can derive the product formula and use it for an arbitrary number of events:
P (A1, A2, A3, …A p)=P (A1|A2…Ap )P(A2|A3…Ap)P (A 3|A4…Ap)… R (Ap-1 |Ap)R (Ap).
Practice
To make it easier to understand how the conditional probability of an event is calculated, let's look at a couple of examples. Suppose there is a vase containing 8 chocolates and 7 mints. They are the same size and random.two of them are pulled out in succession. What are the chances that both of them will be chocolate? Let us introduce notation. Let the result A mean that the first candy is chocolate, the result B is the second chocolate candy. Then you 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 family of two children and we know that at least one child is a girl.
What is the conditional probability that these parents don't have boys yet? As in the previous case, we start with notation. Let P(B) be the probability that there is at least one girl in the family, P(A|B) be the probability that the second child is also a girl, P(AB) be the chances that there are two girls in the family. Now let's do the calculations. In total, there can be 4 different combinations of the sex of children, and in this case, only in one case (when there are two boys in the family), there will be no girl 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.
The result can be interpreted as follows: if we did not know the gender of one of the children, then the chances of two girls would be 25 against 100. But since we know that one child is a girl, the probability that the family of boys no, increases to one-third.
