What is the conditional probability that exactly four heads appear
when a fair coin is flipped five times, given that the first flip came
up tails?

There is a 1/2 chance that a single flip will turn up heads so:
1/2 * 1/2 * 1/2 * 1/2 = 1/16
The fact that the first flip came up as tails in no way affects the
probability of subsequent flips.

Let E be the events of four heads (in 4 flips), and C be the event of
first flip came up tail.
 Here, I use * to mean the intersection, E*C means first flip was tail
and
 followed by 4 heads.
 P(E|C) = P(E* C)/ P(C)
 = (1/2)^5/ (1/2)
 = (1/2)^4
 = 1/16

 Kenny