8 junio, 2024

Additive principle: what it consists of and examples

He additive principle It is a probability counting technique that allows measuring 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, whether air, sea or land. We cannot go to a place using two means of transport simultaneously; we need to choose just one.

The additive principle tells us that the number of ways we have to make this trip will correspond to the sum of each possible alternative (means of transport) that exists to go to the desired place, this will even include means of transport that stop 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 extremely important to know how many ways an event can be carried out.

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Probability

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

A random experiment or phenomenon is an action that does not always yield the same results, even if it is carried out 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 to determine 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 event A will occur is a real number between zero and one; that is, a number belonging to the interval [0,1]. 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 performing 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 covers the case in which the sample space has a finite number of elements.

In this case, the probability of event A occurring will be the number of alternatives that are 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 the elements of the sample space must be equally probable (for example, like an unaltered die, in which the probability of obtaining any of the six numbers is the same).

For example, what is the probability that rolling a dice will result in 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 above, probability measures the frequency with which a certain event will occur. As part of being able to determine this frequency, it is important to know how many ways such an event can be performed. The additive principle allows us to do this calculation in a particular case.

The additive principle establishes the following: If A is an event that has «a» ways of being done, and B is another event that has «b» ways of being done, and if furthermore only A or B can occur and not both at the same time, same time, then the ways of being realized A or B (A∪B) are a+b.

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

Examples of the additive principle

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 carry out a certain activity or event A, and that there are several alternatives for it, say n.

In turn, the first alternative has a1 ways of being done, the second alternative has a 2 ways of being done, and so on, the nth alternative can be done in n ways.

The additive principle states that the event A can be realized in a1+ a2+…+ an ways.

second example

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

Of one there are two colors available, and of the other five colors available. 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 another, not both simultaneously.

To calculate the different ways of carrying 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 will occur, which is denoted by P(A∪B), knowing that A cannot occur simultaneously with B, is given by P(A∪B)= P(A)+ P(B).

third example

What is the probability of getting a 5 on a die roll or heads on a coin toss?

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

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

References

Bellhouse, D.R. (2011). Abraham De Moivre: Setting the Stage for Classical Probability and Its Applications. CRC Press.
Cifuentes, JF (2002). Introduction to the Theory of Probability. Colombian national.
Daston, L. (1995). Classical Probability in the Enlightenment. Princeton University Press.
Johnsonbaugh, R. (2005). Discrete mathematics. Pearson Education.
Larson, H.J. (1978). Introduction to probability theory and statistical inference. Editorial Limusa.
Lutfiyya, LA (2012). Finite and Discrete Math Problem Solver. Research & Education Association Publishers.
Padro, FC (2001). Discrete math. Politec. of Catalonia.
Steiner, E. (2005). Mathematics for applied sciences. reverse.

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