Page 44 - DMTH404_STATISTICS

Statistics

Notes
Example 5: An electronic manufacturer has two lines A and B assembling identical
electronic units. 5% of the units assembled on line A and 10%of those assembled on line B are
defective. All defective units must be reworked at a significant increase in cost. During the last
eight-hour shift, line A produced 200 units while the line B produced 300 units. One unit is
selected at random from the 500 units produced and is found to be defective. What is the probability
that it was assembled (i) on line A, (ii) on line B?
Answer the above questions if the selected unit was found to be non-defective.
Solution.
Let A be the event that the unit is assembled on line A, B be the event that it is assembled on line
B and D be the event that it is defective.
Thus, we can write

2 3 5 10

P A ,    , D/A  and D/B 

  
P
P B
P
5 5 100 100
Further, we have
2 5 1 3 10 3
P
P A  D     and B  D    
5 100 50 5 100 50
The required probabilities are computed form the following table:
A B Total
1 3 4
D 50 50 50
19 27 46
D
50 50 50
Total
20 30
1
50 50
From the above table, we can write

1 50 1 3 50 3


P A/D    , B/D   
P
50 4 4 50 4 4
19 50 19 27 50 27


P  A/D    ,  B/D   
P
50 46 46 50 46 46
3.3 Independence of Events
From the examples discussed in the previous section you know that the conditional probability
P(A 1 H) is, in general, not the same as the unconditional probability P(A). Thus, the knowledge
of H affects the chances of occurrence of A. The following example illustrates this fact more
explicitly.

Example 18: A box has 4 tickets numbered 1, 2,3 and 4. One of these tickets is drawn at
random. Let A = { 1, 2 } be the event that the randomly selected ticket bears the number 1 or 2.
Similarly define B = { 1 }. Then

P(A ) = 1/2, P(B) = 1/4 and P(A  B) = 1/4.

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