Quoth the GRE:

A fair coin is to be tossed 100 times, with each toss resulting in a head or a tail. If H is the total number of heads and T is the total number of tails, which of the following events has the greatest probability?

(A) H = 50

(B) T ≥ 60

(C) 51 ≤ H ≤ 55

(D) H ≥ 48 and T ≥ 48

(E) H ≤ 5 or H ≥ 95

Answer: D

Okay, so this is a binomial experiment, and we can easily translate all of the answer choices to statements about H alone by substituting T = 100 - H.

My inclination is to approximate the binomial by a normal distribution, except I don't have a table of values for the GRE. I can sort of make do by drawing a rough picture of a normal distribution with mean 50 and comparing areas, though. Doing this, I see that (E) is really small and (A) is less than (D) -- those are out. More, (C) is less than (D) because in both cases H ranges over five values, but in (D) the density function over that range is greater. This leaves (B) and (D). Depending on what I've drawn, I guess (D), but I'm not sure. Why must we disqualify (B)?