Page 90 - DMTH404_STATISTICS

Statistics

Notes This definition of probability is also termed as the empirical definition because the probability
of an event is obtained by actual experimentation.

m
Although, it is seldom possible to obtain the limit of the relative frequency, the ratio can be
n
regarded as a good approximation of the probability of an event for large values of n.
This definition also suffers from the following shortcomings :
(i) The conditions of the experiment may not remain identical, particularly when the number
of trials is sufficiently large.
m
(ii) The relative frequency, , may not attain a unique value no matter how large is the total
n
number of trials.
(iii) It may not be possible to repeat an experiment a large number of times.
(iv) Like the classical definition, this definition doesn't lead to any mathematical treatment of
probability.

6.7 Summary of Formulae

1. (a) The number of permutations of n objects taking n at a time are n!

! n
n
(b) The number of permutations of n objects taking r at a time, are P =
r
(n r- )!
(c) The number of permutations of n objects in a circular order are (n - 1)!
(d) The number of permutations of n objects out of which n are alike, n are alike, ......
1 2
! n
n are alike, are
k
n
n 1 !n 2 ! ... !
k
! n
n
(e) The number of combinations of n objects taking r at a time are C =
r
( ! r n r- )!
2. (a) The probability of occurrence of at least one of the two events A and B is given by :
P (A  ) B = P A ( ) P- (A  B ) 1 P A= - (  B ) .
( ) P B+
(b) The probability of occurrence of exactly one of the events A or B is given by :
P (A  B ) ( A+ P  B ) or P (A ) B - P (A ) B
3. (a) The probability of simultaneous occurrence of the two events A and B is given by:
P (A  ) B = P ( ) ( /A .P B A ) or = P ( ) ( /B .P A ) B

(b) If A and B are independent ( P A ) B = P A .P
( ) ( ) B .
6.8 Keywords

Classical: If n is the number of equally likely, mutually exclusive and exhaustive outcomes of
a random experiment out of which m outcomes are favourable to the occurrence of an event A,
then the probability that A occurs, denoted by P(A), is given by :

Number of outcomes favourable to A m
( ) =
P A =
Number of exhaustive outcomes n
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