Page 117 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

Notes
Example 51: Given below are the daily wages (in rupees) of six workers of a factory :
77, 105, 91, 100, 90, 83

If two of these workers are selected at random to serve as representatives, what is the probability
that at least one will have a wage lower than the average?
Solution.

+
77 105 91 100 90 83
+
+
+
+
The average wage X   91
6
Let A be the event that two workers selected at random have their wages greater than or equal
to average wage.
3 C 1
( ) 
 P A 2 
6
C 2 5
1 4
1
Thus, the probability that at least one of the workers has a wage less than the average   
5 5
Example 52: There are two groups of subjects one of which consists of 5 science subjects
and 3 engineering subjects and the other consists of 3 science subjects and 5 engineering subjects.
An unbiased die is cast. If the number 3 or 5 turns up, a subject from the first group is selected at
random otherwise a subject is randomly selected from the second group. Find the probability
that an engineering subject is selected ultimately.
Solution.
Let A be the event that an engineering subject is selected and B be the event that 3 or 5 turns on
the die. The given information can be summarised into symbols, as given below :
1 3 5
)
)
P ( ) B  , P ( /A B  , and P ( /A B 
3 8 8
To find P(A), we write
( ) 
P A P (A  B ) (A+ P  B )  P ( ) ( /B .P A B ) ( ) ( /P B+ .P A B )

1 3 2 5 13
 ´ + ´ 
3 8 3 8 24

Example 53: Find the probability of obtaining two heads in the toss of two unbiased
coins when (a) at least one of the coins shows a head, (b) second coin shows a head.
Solution.
Let A be the event that both coins show heads, B be the event that at least one coin shows a head
and C be the event that second coin shows a head. The sample space and the three events can be
written as:

S = {(H, H), (H, T), (T, H), (T, T)}, A = {(H, H)},
B = {(H, H), (H, T), (T, H)} and C = {(H, H), (T, H)}.
B  b
Further, A  m H Hgr and A C  b m H Hgr

,
,
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