The complement Rule - with examples
This article will be about the definition and examples of the complement rule, with the formula to find it.
¿What is the complement rule?
The complement rule is one of many statistical methods that we use to calculate the probability of an outcome. First of all this method is called the complement rule referring to when we represent a sample space in a Venn diagram, just as a reminder this diagrams are based on circles, where every circle represents an aleatory outcome and the set of all this circles is the whole sample space.
Another aspect important to know is that a circle (in probability this circles represent an outcome) has a name that represents an outcome, for example, if we have an outcome called “A”, everything that does not belongs to the “A” outcome is called the “A” complement (Usually the complement of an outcome is represented with the same letter of the outcome but with a line above it).
Now having this clear, we understand that we use the complement rule when we want to find everything that has nothing to do with a determined outcome, other methods like the sum rule, the multiplication rule, classical probability, etc, (links to this articles at the end of this page) are focused to find the probability of a determined outcome but in the complement rule we find the opposite of that, that is the probability that a determined outcome does not happen.
The complement rule is applied in problems where it is complicated to find the probability of an outcome or a set of outcomes because the amount of outcomes to find is higher than the outcomes that we do not want to find, and in this cases it is easier to find the probability of the opposite outcomes and based on this probability we can find the probability of the outcomes we are looking for, based on the fact that the sum of all the outcomes will have to be equals to 1.
Example to understan the complement rule
To understand better the situations where the complement rule is more recommended to be used we are going to work in an example: as we already know when we roll a dice there are 6 possible outcomes, one for each side of the dice, and this 6 outcomes form the sample space and every side has the same probability of coming out, so, if we want to know the probability that when we roll the dice we get a number lower than 6 ¿how would we calculate that with the complement rule?
Using other methods we would find the probability of the numbers 1, 2, 3, 4 and 5 because this is what we are looking for, but with the complement rule we are going to do the opposite, because it is easier to find the probability of one side, and then we are going to subtract the probability to 1 (that is the sum of the whole probabilities) to find the probability of the other 5 sides.
Perhaps in the previous example we cannot see that much the difference of using the complement rule, but this was just an example to prove how does this method works, but when we have problems where not all the outcomes have the same probability, is then when we can appreciate the benefits of using the complement rule instead o the traditional probabilities.
Formula of the complement rule
- Formula
- p( a ) = 1 - p(a)
Examples of the complement rule
Example 1: Find the probability that when we roll a dice we get a number different than 1 and 6
- First we find p(a) being “a” getting the number 1 and 6
- p(a) = 2 / 6
- p(a) = 0.3333
- now we find p( a )
- p( a ) = 1 - 0.3333
- p( a ) = 0.6666
- p( a ) = 0.6667
- p( a ) = 0.6667 * 100%
- p( a ) = 66.67%
Example 2: A young man goes to buy a new phone model which has a 10 different color collection, but he does not like two of those colors, if when he buys the phone they give him a random color ¿What are the chances of him getting a color that he actually likes?
- First we find p(c) being c getting one of the unwished colors
- p(c) = 2 / 10
- p(c) = 0.2
- Then we find p( c )
- p( c ) = 1 - 0.2
- p( c ) = 0.8
- p( c ) = 0.8 * 100%
- p( c ) = 80%
Example 3: in a company they are making their employees work overtime one day of the next week between Monday and Saturday, but one of the workers has a surprise party for his son´s birthday next Friday, so if this extra hours are given randomly, what is the probability that this worker has to work overtime any day but friday.
- First we find p(o) o=overtime work friday
- p(o) = 1 / 6
- p(o) = 0.1667
- And then we find p( o )
- p( o ) = 1 - 0.1667
- p( o ) = 0.8333
- p( o ) = 0.8333 * 100%
- p( o ) = 83.33%
Example 4: In an ice cream shop there are 4 types of flavors, according to the statistics of the shop, the favorite flavor of its clients is the chocolate flavor, and because of this the chocolate flavor is running out of stock and the people from the ice cream shop do not want people to ask for this flavor because they would be out of chocolate too fast, knowing this what is the probability that the next client asks for any flavor but chocolate.
- we find p(c) being c the cocolate flavor
- p(c) = 1 / 4
- p(c) = 0.25
- And we find p( c )
- p( c ) = 1 - 0.25
- p( c ) = 0.75
- p( c ) = 0.75 * 100%
- p( c ) = 75%
Example 5: A teacher is asking to some students to read for the class the lesson of the day, if the teacher is going to put to read to 7 of his 40 students, but most of the students are very shy to read for the class ¿What are the chances for each student to not read out loud?
- First we find p(r) where r is reading for the class
- p(r) = 7 / 40
- p(r) = 0.175
- And finaly we find p( r )
- p( r ) = 1 - 0.175
- p( r ) = 0.825
- p( r ) = 0.825 * 100%
- p( r ) = 82.5%