Chapter 15.7, Problem 10P

Show that the most probable number of heads in $n$ tosses of a coin is $\frac{1}{2}n$ for even $n$ [that is, $f(x)$ in (7.1) has its largest value for $x=n/2]$ and that for odd $n$, there are two equal "largest" values of $f(x),$ namely for $x=\frac{1}{2}(n+1)$ and $x=\frac{1}{2}(n-1)$ Hint: Simplify the fraction $f(x+1)/f(x),$ and then find the values of $x$ for which it is greater than 1 [that is, $\text{f}(\text{x+1})>\text{f}(\text{x})]$, and less than or equal to 1 [that is, $f(x+1)\le f(x)].$ Remember that $x$ must be an integer.