What happens if, on inequality, you have a 50% chance of doing one more comparison operation?

The way I figure, which may be incorrect, is the following:

You have x options (here it is 16). x^2 gives us 256 different options for this example. However, if it's only correct correct half of the time then we have to repeatedly cut down our search which results in series:

sum (x^2)/(2n), n=1 to m

which is

(x^2 H_m) / 2

Is this correct? Could someone explain how many random extra comparisons would be needed to thwart a timing attack?