You have

.

If you draw a graph of

and draw upper and lower rectangles over the appropriate partition it should be clear that

Substitute

:

.

Therefore, for large n and therefore large enough r:

.

where the substitution

is made.

Note that r-1 is the number of applicants initially rejected under the strategy and n is the total number of applicants. x therefore gives the proportion of applicants intially rejected.

All that's left is to find the value of x that maximises Pr(r) in this limit of large n ......