Statistical Methods

1.6.1.Definition of probability - Definition-II

Unit 1 - Basic concepts of probability

1.6.1.Definition-II : Relative frequency or aposteriori or empirical definition of probability
Classical definition of probability defined earlier assumes that outcomes are equally likely the total number of outcomes is known finite. When these assumptions are not met with, it is not possible to compute the probability of an event using the classical definition. In order to overcome the above limitations, a new approach called relative frequency concept of probability is adopted. According to this concept the probability of occurrence of an event E is the limiting value of ratio of frequency of occurrence of the event to the total number of outcomes. For instance, if an experiment was repeated n times under the same conditions and if an event E has occurred f times, then the estimate of the probability of an event E as the number of trials n Increase indefinitely is given by,

It is to be noted that as the number of trials (frequency) increases the estimate of probability of an event stabilizes around a particular value.

Example 5
The frequency distribution of lengths of 1000 randomly selected fishes of a particular species is given below. What is the probability that a fish chosen at random will have length between 35-45 cm?

Answer

• Frequency of the class interval 35-45 is 220.
• Therefore, the relative frequency of this class to the total frequency is 220/1000 = 0.22.
• Hence, probability that the fish chosen at random will have length between 35-45 cm is 0.22.

Example 6
One thousand fertilized eggs of a major carp were kept under observation to find out the number of individuals reaching different stages in the life history. Observed data are given below:

Find the probability that,

• Fertilized egg reaches fingerling stage
• Hatchling reaches fry stage
• Fry reaches adult stage

Answer

• Out of 1000 fertilized eggs, only 200 reached the fingerlings stage. Therefore, the probability of fertilized I egg reaching the fingerling stage is 200/1000 = 0.2
• Out of 700 hatchlings only 210 reached the fry stage, therefore the probability of hatchling reaching the fry stage is 210/700 = 0.30
• Out of 210 fry 196 reached the adult stage, therefore, the probability of the fry reaching the adult stage = 196/210 = 0.928