# Frequency probability

The frequency or frequentist probability refers to the definition of probability understood as the quotient between the number of favorable cases and the number of possible cases, when the number of cases tends to infinity.

Mathematically the frequency probability is expressed as:

Where:

s: is a specific event

N: Total number of events

P (s): It is the probability of the event s

Intuitively this is read as the limit of the frequency as n approaches infinity. In simple words, the value to which the probability of an event tends, when we repeat the experiment many times.

For example, a coin. If you flip a coin 100 times, it can come up 40 times heads and 60 times tails. Of course, this result (which could have been any other) does not indicate that the probability of heads is 40% and the probability of tails is 60%. No. What the frequency probability tells us is that when we flip the coin infinitely many times the probability should stabilize at 0.5. As long as, of course, the coin is perfect.

## Properties of the definition of frequency probability

The frequentist or frequency definition of probability has characteristics that are worth mentioning. The properties are:

• The probability of an event S will always be between 0 and 1.

Indeed, we can demonstrate this fact, using the formula above. On the one hand, we know that the event S will always be less than the total number of trials. It is logical to think that if we repeat the experiment N times, the maximum number of times that S will occur will be equal to N. Thus:

That is, starting from the premise explained above, we divide (second step) all elements by N. Once this is done, we reach the conclusion surrounded in red. That is, the frequency probability or relative frequency of an event will always be between 0 and 1.

• If an event S is the union of a set of disjoint events, its probability is equal to the sum of the probabilities of each separate event.

Two disjoint events are those that do not have elementary events in common. Therefore, it makes sense to think that the probability of an event (S) that is the result of the sum of relative frequencies of each event (s). Mathematically it is expressed like this:

In the previous operation it is translated from absolute frequencies to relative frequencies. That is, understood S as a set of disjoint events (s), its union is equal to the sum of all of them. This would give us the absolute frequency as the result. That is, the total number of times the event occurs. To convert it to probability, we only have to divide this number by N. Or, even better, add the probabilities of each event (s) that make up event S.

See relationship between absolute and relative frequency

## Criticisms of the definition of frequency probability

As one might expect, the definition of frequency or frequency probability was born a few years ago. Specifically, around the year 1850 the concept began to develop. However, it would not be until 1919 when it would be formally developed by Von Mises. The Austrian economist based his theory of frequency probability on two premises:

• Statistical regularity: Although the behavior of the specific results is somewhat chaotic, after repeating an experiment a large number of times, we find certain patterns of results.
• Probability is an objective measure: Von Mises argued that probability could be measured and, furthermore, it was objective. To defend this argument, he relied on the fact that random phenomena have certain characteristics that make them unique. Derived from the above, we can understand its repetition patterns.

Taking into account the above, and despite the fact that the concept of frequency probability is postulated as the only empirical way to calculate probabilities, the concept has received the following criticisms:

• The concept of limit is unreal: The formula proposed for the concept assumes that the probability of an event must stabilize when we repeat the experiment infinitely many times. That is, when N tends to infinity. However, in practice it is impossible to repeat something infinitely many times.
• It does not assume a truly random sequence: The concept of limit, at the same time, assumes that a probability must stabilize.However, the very fact of stabilizing, mathematically, does not allow us to assume that the sequence is truly random. In some way, it indicates that it is something specific.
Cumulative absolute frequency

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