Question

40 numbers are rounded off to the nearest integer and then summed. If the individual round-off error are uniformly distributed over (−.5,.5) what is the probability that the resultant sum differs from the exact sum by more than 2 ?

Answer 1

Let the error of individual round off be denoted by W_i.

Here

If the original numbers are X_i and the numbers to which they are rounded off are Y_i,

then, X_i=Y_i + W_i.

It is given that  

Hence

Hence by Central limit theorem

The exact sum is given by

And sum of rounded off numbers is given by

Therefore the sum of errors will be given by

Note that

We want the probability that the resultant sum differs from the exact sum by more than 2.

Which is

P[The resultant sum differs from the exact sum by more than 2]

P[The resultant sum differs from the exact sum by more than 2] = 0.2734