That's a very interesting question.
Here's the way I thought about it, and the answer I came up with
while avoiding the need to define and count every possible outcome:
-- There are (2 x 2 x 2 x 2) = 16 possible outcomes of 4 flips.
-- There are several different ways to get 1 heads and 3 tails.
-- There are several different ways to get 2 heads and 2 tails.
-- There are several different ways to get 3 heads and 1 tails.
-- But there's only one way to get heads all 4 times,
and there's only one way to get tails all 4 times.
-- So there are 2 ways for all 4 flips to produce the same face.
The probability of that is 2/16 = 1/8 = 12.5% , and I'm saying
that this is the least probable result of 4 flips.
