Statistical Methods

Unit 1 - Basic concepts of probability

1.6.Definition of Probability
We present here two definitions of probability.

Definition I - Classical or mathematical or apriority definition
Suppose an event E can happen in 'm' different ways (outcomes) out of a total of 'n' different exhaustive, mutually exclusive and equally likely ways, then the probability of occurrence of an event denoted by P (E) is given by,

Note 1
Probability of an event is a non-negative number which lies between 0 and 1.
Symbolically 0 ≤ p ≤ 1.
Note 2
If the event E can happen in 'm' ways out of total of ‘n’ ways, then the number of ways in which the event A will not happen is n - m. Hence, the probability that an event E will not happen (denoted by q) is given by,

So that, p+q=1
i.e., the sum of the probabilities of occurrence and non-occurrence of an event is equal to 1,

Example 1
What is the probability of getting head when an unbiased coin is tossed ?
Answer:
Total number of equally likely outcomes = 2 = { H, T}
No. of favorable outcome =1 = {H }
Therefore, the probability of getting head is, P (H) = ½.
Example 2
What is the probability of getting an even number when an unbiased die is thrown?
Answer
Total Number of equally likely outcomes = 6
Number of favorable outcomes
P (Even number) =3/6=0.5
Example 3
In a pond containing 100 fishes 20 are marked. If one fish is subsequently caught what is the probability of it being (i) marked (ii) unmarked?
Answer
Total number of fishes =100
Number of marked fishes =20

(i) Hence the numbers of favorable cases for marked fish are 20.

(ii) P(marked fish being caught) + P (unmarked fish being caught) = 1
Hence, P(unmarked fish being caught) = 1- P(marked fish being caught) = 1-0.2 = 0.8

Example 4
In a composite fish culture experiment, fingerlings of 6 species of fish namely, rohu, catla, mrigal, common carp, silver carp and grass carp, were stocked in the ratio of 1 : 1 : 1 : 2.5 : 3 : 1.5 respectively. A fingerling is subsequently drawn, what is the probability that it is of catla?
Answer
Fingerlings of rohu, catla, mrigal, common carp, silver carp and grass carp are stocked in the ratio of 1 : 1 : 1 : 2.5 : 3 : 1.5 respectively. Thus out of 10 fingerlings we have 1 fingerling of catla. Hence, the probability that the fingerling drawn is of catla, = 1/10 = 0.10