Question

In a railway reservation office, two clerks are engaged in checking rese
rvation forms. On
an average, the first clerk (A
1
) checks 55 per cent of the forms, while the second (A
2
)
checks the remaining. While A
1
has an error rate of 0.03 that of A
2
is 0.02. A reservation
form is selected at random from the total number of forms c
hecked during a day and is
discovered to have an error. Find the probabilities that it was checked by A
1
, and A
2
,
respectively
.

Answer 1

Probability of a given form having been checked by A1 = 55% = 0.55
Probability of a given form having been checked by A2 = 100 - 55 = 45 % = 0.45

Probability of A1 making an error in checking a form = 0.03
Probability of A1 checking correctly a given form = 1 - 0.03 = 0.97

Probability of A2 making an error in checking a form = 0.02
Probability of checking correctly a given form = 1 - 0.02 = 0.98

We assume that the two events of checking forms and making errors are independent. So we can multiply their probabilities.

Probability of A1 checking a form and making an error in that form =
       =  probability of checking a form * probability of making an error
       =  0.55 * 0.03  = 0.0165

Probability of A2 checking a form and making an error in that form =
       =  probability of checking a form * probability of making an error
       =  0.45 * 0.02  = 0.0090

Total probability of a form being checked and erroneously checked = 
        = 0.0165 + 0.0090 = 0.0255

One form, that was checked already, has an error. 

Probability of A1 having checked that particular form =
                     0.0165 / 0.0255 = 0.647 or 64.7%

Probability of A2 having checked that particular form = 
                       = 0.0090 / 0.0255 = 0.353 or 35.3 %