WHAT IS CLASSICAL PROBABILITY? (WITH SOLVED EXERCISES) - MATHS - 2025

The classical probability is a particular case of calculating the probability of an event. To understand this concept it is necessary to first understand what the probability of an event is.

Probability measures how likely an event is to happen or not. The probability of any event is a real number that is between 0 and 1, inclusive.

If the probability of an event happening is 0 it means that it is certain that that event will not happen.

On the contrary, if the probability of an event happening is 1, then it is 100% certain that the event will happen.

Probability of an event

It was already mentioned that the probability of an event happening is a number between 0 and 1. If the number is close to zero, it means that the event is unlikely to happen.

Equivalently, if the number is close to 1 then the event is quite likely to happen.

Also, the probability that an event will happen plus the probability that an event will not happen is always equal to 1.

How is the probability of an event calculated?

First the event and all possible cases are defined, then the favorable cases are counted; that is to say, the cases that are of interest to happen.

The probability of this event "P (E)" is equal to the number of favorable cases (CF), divided by all possible cases (CP). That is to say:

P (E) = CF / CP

For example, you have a coin such that the sides of the coin are heads and tails. The event is to flip the coin and the result is heads.

Since the coin has two possible outcomes but only one of them is favorable, then the probability that when the coin is tossed the outcome will be heads is equal to 1/2.

Classical probability

The classical probability is one in which all possible cases of an event have the same probability of occurring.

According to the previous definition, the event of a coin toss is an example of classical probability, since the probability that the result is heads or tails is equal to 1/2.

The 3 most representative classical probability exercises

First exercise

In a box there is a blue, a green, a red, a yellow and a black ball. What is the probability that, when removing a ball from the box with closed eyes, it will be yellow?

Solution

The event "E" is to remove a ball from the box with the eyes closed (if it is done with the eyes open the probability is 1) and that it is yellow.

There is only one favorable case, since there is only one yellow ball. The possible cases are 5, since there are 5 balls in the box.

Therefore, the probability of event "E" is equal to P (E) = 1/5.

As can be seen, if the event is to draw a blue, green, red or black ball, the probability will also be equal to 1/5. So this is an example of classical probability.

Observation

If there had been 2 yellow balls in the box then P (E) = 2/6 = 1/3, while the probability of drawing a blue, green, red or black ball would have been equal to 1/6.

Since not all events have the same probability, then this is not an example of classical probability.

Second Exercise

What is the probability that, when rolling a die, the result obtained is equal to 5?

Solution

A die has 6 faces, each with a different number (1,2,3,4,5,6). Therefore, there are 6 possible cases and only one case is favorable.

So, the probability that rolling the die will get 5 is equal to 1/6.

Again, the probability of getting any other roll on the die is also 1/6.

Third Exercise

In a classroom there are 8 boys and 8 girls. If the teacher randomly selects a student from her classroom, what is the probability that the student chosen is a girl?

Solution

Event "E" is randomly choosing a student. In total there are 16 students, but since you want to choose a girl, then there are 8 favorable cases. Therefore P (E) = 8/16 = 1/2.

Also in this example, the probability of choosing a child is 8/16 = 1/2.

In other words, the chosen student is as likely to be a girl as it is a boy.

References

1. Bellhouse, DR (2011). Abraham De Moivre: Setting the Stage for Classical Probability and Its Applications. CRC Press.
2. Cifuentes, JF (2002). Introduction to the Theory of Probability. National University of Colombia.
3. Daston, L. (1995). Classical Probability in the Enlightenment. Princeton University Press.
4. Larson, HJ (1978). Introduction to probability theory and statistical inference. Editorial Limusa.
5. Martel, PJ, & Vegas, FJ (1996). Probability and mathematical statistics: applications in clinical practice and health management. Díaz de Santos editions.
6. Vázquez, AL, & Ortiz, FJ (2005). Statistical methods to measure, describe and control variability. Ed. University of Cantabria.
7. Vázquez, SG (2009). Manual of Mathematics for access to the University. Editorial Centro de Estudios Ramon Areces SA.