AppendixB 185Hence lnP=0,orP= 1 ,withinthe accuracyofStirling’s approximation. Another
way of stating this result is as:t∗=,a result we have used in Chapter 2 and
elsewhere. Since wehave notedthe success of(B.1) we canhave confidenceinour
statisticalmethod.
However, before leaving this example, a word of warning. It is not in fact true
that the probability ofgettingexactlyN/ 2 HandN/ 2 Tincreases towards1asN
increases!RatherlnPgets smaller aslnN(the errorin (B.1)), as maybe verified
by anyone with a laptop computer and a sense of adventure. But the sharpness of
the probabilityat the maximumdoesincrease asNgetslarger,definedsaybythe
probabilityofgettinga distribution within 0.1% of the most probable. In other words it
is the probability of gettingnearlythe most probable which becomes overwhelmingly
large, on anyreasonabledefinition of‘nearly’. Fortunately,thisis preciselythe sort
of situation in which we are interested in statistical physics.