Page 89 - DMTH404_STATISTICS

Unit 6: Probability

Notes
8! 1 280
 The required probability =  = .
3!4!1! 3 8 6561

Example 22: 12 'one rupee' coins are distributed at random among 5 beggars A, B, C, D
and E. Find the probability that :
(i) They get 4, 2, 0, 5 and 1 coins respectively.
(ii) Each beggar gets at least two coins.

(iii) None of them goes empty handed.
Solution.
The total number of ways of distributing 12 one rupee coins among 5 beggars are
+
-
12 5 1 16
C = C = 1820 .
-
5 1 4
(i) Since the distribution 4, 2, 0, 5, 1 is one way out of 1820 ways, the required probability
1
= .
1820
(ii) After distributing two coins to each of the five beggars, we are left with two coins, which
-
+
can be distributed among five beggars in 2 5 1 C = 6 C = 15 ways.
-
5 1 4
15 3
 The required probability = = .
1820 364
(iii) No beggar goes empty handed if each gets at least one coin. 7 coins, that are left after
giving one coin to each of the five beggars, can be distributed among five beggars in
-
+
7 5 1 11
C 5 1 = C = 330 ways.
4
-
330 33
 The required probability = = .
1820 182
6.3 Statistical or Empirical definition of Probability
The scope of the classical definition was found to be very limited as it failed to determine the
probabilities of certain events in the following circumstances :
(i) When n, the exhaustive outcomes of a random experiment is infinite.
(ii) When actual value of n is not known.

(iii) When various outcomes of a random experiment are not equally likely.
In addition to the above this definition doesn't lead to any mathematical treatment of probability.
In view of the above shortcomings of the classical definition, an attempt was made to establish
a correspondence between relative frequency and the probability of an event when the total
number of trials become sufficiently large.

6.3.1 Definition (R. Von Mises)

If an experiment is repeated n times, under essentially the identical conditions and, if, out of
these trials, an event A occurs m times, then the probability that A occurs is given by P(A) =
, provided the limit exists.

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