Page 88 - DMTH404_STATISTICS

Statistics

Notes
Example 19: 5 red and 2 black balls, each of different sizes, are randomly laid down in a
row. Find the probability that
(i) the two end balls are black,
(ii) there are three red balls between two black balls and
(iii) the two black balls are placed side by side.

Solution.
The seven balls can be placed in a row in 7! ways.
(i) The black can be placed at the ends in 2! ways and, in-between them, 5 red balls can be
placed in 5! ways.

2!5! 1
 The required probability = = .
7! 21
(ii) We can treat BRRRB as one ball. Therefore, this ball along with the remaining two balls
can be arranged in 3! ways. The sequence BRRRB can be arranged in 2! 3! ways and the three
5
red balls of the sequence can be obtained from 5 balls in C ways.
3
3!2!3! 1
 The required probability =  5 C = .
7! 3 7
(iii) The 2 black balls can be treated as one and, therefore, this ball along with 5 red balls can be
arranged in 6! ways. Further, 2 black ball can be arranged in 2! ways.

6!2! 2
 The required probability = = .
7! 7

Example 20: Each of the two players, A and B, get 26 cards at random. Find the probability
that each player has an equal number of red and black cards.
Solution.
52
Each player can get 26 cards at random in C ways.
26
In order that a player gets an equal number of red and black cards, he should have 13 cards of
each colour, note that there are 26 red cards and 26 black cards in a pack of playing cards. This can
26 C  26 C
26
be done in C  26 C ways. Hence, the required probability = 13 13 .
13
13
52
C 26
Example 21: 8 distinguishable marbles are distributed at random into 3 boxes marked as
1, 2 and 3. Find the probability that they contain 3, 4 and 1 marbles respectively.
Solution.
Since the first, second .... 8th marble, each, can go to any of the three boxes in 3 ways, the total
8
number of ways of putting 8 distinguishable marbles into three boxes is 3 .
The number of ways of putting the marbles, so that the first box contains 3 marbles, second
8!
contains 4 and the third contains 1, are
3!4!1!

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