ADDITIVE PRINCIPLE: WHAT IT CONSISTS OF AND EXAMPLES - MATHS - 2025

The additive principle is a probability counting technique that allows us to measure in how many ways an activity can be carried out, which, in turn, has several alternatives to be carried out, of which only one can be chosen at a time. A classic example of this is when you want to choose a transport line to go from one place to another.

In this example, the alternatives will correspond to all the possible transport lines that cover the desired route, either air, sea or land. We cannot go to a place using two means of transport simultaneously; we need to choose only one.

The additive principle tells us that the number of ways we have to make this trip will correspond to the sum of each alternative (means of transport) possible that exists to go to the desired place, this will even include the means of transport that make a stopover somewhere (or places) in between.

Obviously, in the previous example we will always choose the most comfortable alternative that best suits our possibilities, but probabilistically it is very important to know in how many ways an event can be carried out.

Probability

In general, probability is the field of mathematics that is responsible for studying events or phenomena and random experiments.

An experiment or random phenomenon is an action that does not always yield the same results, even if it is performed with the same initial conditions, without altering anything in the initial procedure.

A classic and simple example to understand what a random experiment consists of is the action of tossing a coin or a dice. The action will always be the same, but we will not always get "heads" or a "six", for example.

Probability is responsible for providing techniques for determining how often a given random event can occur; among other intentions, the main one is to predict possible future events that are uncertain.

Probability of an event

More particularly, the probability that an event A occurs is a real number between zero and one; that is, a number belonging to the interval. It is denoted by P (A).

If P (A) = 1, then the probability of event A occurring is 100%, and if it is zero there is no chance of it occurring. The sample space is the set of all possible outcomes that can be obtained by conducting a random experiment.

There are at least four types or concepts of probability, depending on the case: classical probability, frequentist probability, subjective probability and axiomatic probability. Each one focuses on different cases.

Classical probability encompasses the case in which the sample space has a finite number of elements.

In this case, the probability of an event A occurring will be the number of alternatives available to obtain the desired result (that is, the number of elements in set A), divided by the number of elements in the sample space.

Here it must be considered that all elements of the sample space must be equally probable (for example, as a given that is not altered, in which the probability of obtaining any of the six numbers is the same).

For example, what is the probability that rolling a die will get an odd number? In this case, the set A would be made up of all the odd numbers between 1 and 6, and the sample space would be made up of all the numbers from 1 to 6. So, A has 3 elements and the sample space has 6. So Therefore, P (A) = 3/6 = 1/2.

What is the additive principle?

As stated earlier, probability measures how often a certain event occurs. As part of being able to determine this frequency, it is important to know in how many ways this event can be carried out. The additive principle allows us to make this calculation in a particular case.

The additive principle establishes the following: If A is an event that has “a” ways of being performed, and B is another event that has “b” ways of being performed, and if in addition only A or B can occur and not both at the same time, then the ways to be realized A or B (A deB) are a + b.

In general, this is stated for the union of a finite number of sets (greater than or equal to 2).

Examples

First example

If a bookstore sells books on literature, biology, medicine, architecture, and chemistry, of which it has 15 different types of books on literature, 25 on biology, 12 on medicine, 8 on architecture, and 10 on chemistry, how many options does a person have to choose an architecture book or a biology book?

The additive principle tells us that the number of options or ways to make this choice is 8 + 25 = 33.

This principle can also be applied in the event that a single event is involved, which in turn has different alternatives to be carried out.

Suppose that you want to perform a certain activity or event A, and that there are several alternatives for it, say n.

In turn, the first alternative has ₁ ways of being done, the second alternative has ₂ ways of being done, and so on, alternative number n can be done in _n ways.

The additive principle states that event A can be performed in ₁ + to ₂ +… + in _n ways.

Second example

Suppose a person wants to buy a pair of shoes. When she arrives at the shoe store, she finds only two different models of her shoe size.

There are two available colors of one, and five available colors of the other. How many ways does this person have to make this purchase? By the additive principle the answer is 2 + 5 = 7.

The additive principle should be used when you want to calculate the way to perform one event or the other, not both simultaneously.

To calculate the different ways to carry out an event together ("and") with another - that is, that both events must occur simultaneously - the multiplicative principle is used.

The additive principle can also be interpreted in terms of probability as follows: the probability that an event A or an event B occurs, which is denoted by P (A∪B), knowing that A cannot occur simultaneously to B, is given by P (A∪B) = P (A) + P (B).

Third example

What is the probability of getting a 5 when rolling a die or heads when tossing a coin?

As seen above, in general the probability of getting any number when rolling a die is 1/6.

In particular, the probability of getting a 5 is also 1/6. Similarly, the probability of getting heads when tossing a coin is 1/2. Therefore, the answer to the previous question is P (A∪B) = 1/6 + 1/2 = 2/3.

References

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