This question appeared on my GRE practice test, and I am wondering if there is a good general strategy for questions like this:
A fair coin is flipped 100 times, yielding $H$ heads and $T$ tails. Which of the following is most likely?
A) $H=50$
B) $T>60$
C) $48 \leq T$ and $48 \leq H$
D) $51 \leq T \leq 55$
E) $H\leq 5$ or $H \geq 95$
I eliminated A because it is a subset of C, and E because the tails of the distribution will be very very small. I eliminate D because C also has five events which are overall more likely. How do i compare $B$ and $C$?
I know how to solve this problem by brute force, but I am looking for a way to solve it in the 2 minutes 34 seconds allowed for each question on this test!
With the modification, the only reasonable choice is (C). It is pretty big, we can think of it as the probability of being within $0.4$ to $0.5$ standard deviation units of the mean.
For comparison with (D), note that the two intervals have the same length but (C) is "central." Recall that the binomial coefficients reach a maximum at the middle, and are symmetrical, reaching, in this case, a maximum at $50$. So any interval of length $k$ symmetrical about the mean has greater probability than any non-symmetrical interval of the same length.
As to (B), it is more than $2$ standard deviation units up from the mean. The choices (A) and $(E)$ are ruled out for the reasons you described.