Tabla de Contenidos
In statistics and probability, the complement rule establishes that the probability that any event A will occur will always be equal to unity minus the probability that the opposite or complementary event to A will occur . In other words, it is a rule that indicates that the probabilities of an event and its complement are related by means of the following expression:
This rule is one of the basic properties of probability and tells us that we can always calculate the probability of any event if we know the probability of its complement and vice versa. This is particularly important, since in many real-world situations where we need to compute the probability of an event it is much easier to compute the probability of its complement directly instead. Then, once this is calculated, we use the complement rule to determine the probability that we wanted initially.
Some simple examples of the application of this rule are:
- If the probability that Real Madrid will win a Champions League soccer match is 34/57 or 0.5965, the probability that they will not win a Champions League match is 1-34/57 = 23/57 or 0.4035.
- The probability that a common 6-sided die will land on an even number less than 6 is 1/3, so the probability that the die will not land on an even number less than 6 is 2/3.
Proof of the complement rule
The complement rule can be demonstrated in several different ways, any of which will make it easier for the reader to remember. In order to do this demonstration, we must start by defining some basic terms such as what is an event and what is its complement. In addition, we must state some of the main axioms on which probability is based.
Experiments, results, sample space and events
In statistics and probability we talk about carrying out experiments , such as flipping coins, rolling a dice, choosing a card or deck from a randomly shuffled deck, and so on. Every time we carry out an experiment, we get a result , such as choosing the 2 of clubs from the deck of Spanish playing cards.
The total set of all possible different results that an experiment can give is called the sample space and is usually represented by the letter S.
On the other hand, a particular result or set of results of the experiment is known as an event . Events can be individual results, in which case they are called simple events, or they can be compound events that are made up of more than one element or result.
What is the plugin of an event?
The complement of an event is nothing more than the set of all other possible outcomes in the sample space that do not include the outcomes of the event itself . In the case of the example of rolling a die, the complement of the event in which the die lands on 5, for example, is another event in which the die lands on 1, 2, 3, 4, or 6, or whatever. It’s the same, it doesn’t fall in 5.
Plugins are often represented in different ways. The two most common ways are:
- Placing a slash above the event name (for example, A̅ represents the complement of event A).
- Placing a C as superscript (A C ).
In either case, it reads “A-complement,” “complement of A,” or “Not A.”
An easy way to understand both the plugin concept and the plugin rule itself is by using Venn diagrams . The following figure shows a simple diagram of any experiment and a single event that we will call A.
In Venn diagrams like this one, the entire rectangle represents the sample space of the experiment, while the entire area of the rectangle (in this case, both the gray and blue areas) represents the probability of the sample space, which, by definition , is equal to 1. This is because, if we carry out an experiment, it is absolutely certain that some result contained in the sample space will be obtained, since it contains all the possible results.
The blue circle encloses the area of show space in which all possible outcomes of event A are supposed to lie. For example, if event A is rolling an even number, then this blue area must contain a the results 2, 4 and 6. On the other hand, all the area that is outside the event A (that is, the gray zone), is the complement of A since it contains the other results (1, 3 and 5). .
The complement rule and Venn diagrams
A key to understanding the complement rule using Venn diagrams is that the area of any event within these diagrams is proportional to its probability; the total area of the rectangle corresponds to a probability of 1. As we can clearly see, the event A (blue circle) and its complement, A̅ (gray area) together form the entire rectangle.
For this reason, the sum of their areas, which represent their respective probabilities, must be equal to 1, which is the area of the sample space, S. Rearranging this, we would obtain:
This is the complement rule.
The complement rule from the axioms of probability
Any event and its complement form a pair of disjoint or mutually exclusive events, since if one happens, it is impossible, by definition, for the other to happen. Under these conditions, the union probability of these two events is simply given by the sum of the individual probabilities. That is to say:
Also, as we said before, the union of events A and its complement, A C , results in the sample space:
Substituting P(AUC C ) into the above equation and then substituting the probability of S which by definition is 1, we get:
Rearranging the last two members we obtain the complement rule.
Example of a plugin rule application problem
The following is an example of a typical problem where the use of the plugin rule is particularly useful.
statement
Suppose we have a circuit made up of 5 identical chips connected in series, that is, one after the other. The probability that a chip will fail within the first year of its manufacture is 0.0002. If any one of the 5 chips fails, the entire system fails. You want to find the probability that the system will fail in the first year.
Solution
Let us call F (for failure) the result in which a component or system chip fails and E (success) for the result in which the component does not fail or, what is the same, it does work. Then, the data provided by the statement is:
The experiment in which it is determined whether the entire system fails actually corresponds to carrying out 5 simultaneous experiments in which it is determined whether any of the components fail. So, the sample space for this experiment consists of all combinations of success or failure outcomes on each of the 5 components. Being connected in series, we know that the order does matter. Therefore, the sample space is formed by:
This sample space contains 2 5 =32 possible outcomes corresponding to all possible combinations of Es and Fs. Since we want to calculate the probability that the system fails, the event we are interested in, which we will call event A, is given by all outcomes in which at least one of the components fails. In other words, it is given by the following result set:
In fact, there are 2 5 -1=31 possible outcomes in which at least one of the five components fails. If we wanted to calculate the probability of A (that is, P(A)), we would need to calculate the probability of each of these outcomes; it would be considerable work.
However, let us now consider the complementary event of A, that is, the event in which the system does work (which we will call A C ). As we can see, the only way for the whole system to work is for all five components of the circuit to work, that is:
Calculating this probability is much easier than calculating the previous one. Then, given this probability, we use the complement rule to calculate the probability of A. Since the outcomes of each chip are independent events of one another, the probability of A C is simply the product of the probability that each chip works, is say:
But what is the probability of E? Remember that each chip either works or doesn’t work, so E is the complement of F. Therefore, if we have the probability of F (which is given in the exercise), we can calculate the probability of E using the complement rule :
Now we can calculate the probability that the complete system works:
And, again applying the complement rule, we calculate the probability that the system fails:
Answer
The probability that the system will fail in the first year is 0.010 or 1.0%.
References
Devore, JL (1998). PROBABILITY AND STATISTICS FOR ENGINEERING AND SCIENCES . International Thomson Publishers, SA
Complement Rule . (nd). Fhybea. https://www.fhybea.com/complement-rule.html
Rule of the complement in probabilities . (2021, January 1). MateMobile. https://matemovil.com/regla-del-complemento-en-probabilidades/
