In a railway reservation
office, two clerks are engaged in checking reservation forms. On an average,
the first clerk (A1) checks 55 per cent of the
forms, while the second (A2) checks the remaining. While A1 has an error
rate of 0.03 that of A2 is 0.02. A reservation form is selected at random from
the total number of forms checked during a day and is discovered to have an
error. Find the probabilities that it was checked by A1, and A2, respectively.
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 %