Relative frequency

The relative frequency (statistical probability) of an event is the ratio of the number of trials, in which the event has appeared, to the total number of actually made trials.

Thus, the relative frequency of the event A is defined by the formula:

where m is the number of appearances of the event, n is the total number of trials.

Comparing the definitions of probability and relative frequency, we conclude: the definition of probability does not demand that the trials should be made actually; the definition of relative frequency assumes that the trials were made actually. In other words, the probability is calculated before an experiment, and the relative frequency – after an experiment.

Example. The quality department has detected 3 non-standard details in a group consisting of 80 randomly selected details. The relative frequency of appearance of non-standard details is W(A) = 3/80.

Example. There have been made 24 shots in a target, and 19 hits were registered. The relative frequency of hit in the target is W(A) = 19/24.

The long observations have shown that if experiments are made in identical conditions, in each of which the number of trials is rather great, the relative frequency has a stability property. This property is that for different experiments the relative frequency is changed a little (the less changes, the more trials were made), oscillating about some constant number. There was found out that this constant number is the probability of appearance of the event.

Thus, if the relative frequency is established by a practical experiment, the obtained number can be accepted for approximate value of probability.

Geometric probabilities

To overcome defect of the classical definition of probability consisting that it is inapplicable to trials with infinite number of events (outcomes) enter geometric probabilities – the probability of hit of a point in area (segment, part of a plane and etc.).

Let a segment l be a part of a segment L. A point is set (thrown) at random in the segment L. It means that the following suppositions hold: the thrown point can appear in any point of the segment L, the probability of hit of the point in the segment l is proportional to the length of this segment and does not depend on its disposition concerning the segment L. In these suppositions the probability of hit of the point in the segment l is determined by the equality P = the length of l / the length of L.

Example. A point B(x) is thrown at random in a segment OA of the length L of the numeric axis Ox. Find the probability that the smaller of the segments OB and BA has the length more than L/3. It is assumed that the probability of hit of a point in the segment is proportional to the length of the segment and does not depend on its disposition on the numeric axis.

Solution: Let's divide the segment OA by points C and D on three equal parts. The request of the problem will be executed if the point B(x) will hit in the segment CD of the length L/3. The required probability P = (L/3)/L = 1/3.

Let a flat figure g be a part of a flat figure G. A point is thrown at random in the figure G. It means that the following suppositions hold: the thrown point can appear in any point of the figure G, the probability of hit of the thrown point in the figure g is proportional to the area of this figure and does not depend on both its disposition concerning the figure G and the form of g. In these suppositions the probability of hit of the point in the figure g is determined by the equality P = the area of g / the area of G.

Example. Two concentric circles of which the radiuses are 5 and 10 cm respectively are drawn on the plane. Find the probability that the point thrown at random in the large circle will hit in the ring formed by the constructed circles. It is assumed that the probability of hit of a point in a flat figure is proportional to the area of this figure and does not depend on its disposition concerning the large circle.

Solution: The area of the ring (the figure g)

The area of the large circle (the figure G)

The required probability