As various commenters have pointed out, the outcome of a single flip of a coin is either "heads" or "tails." So far, no problem. But seeming difficulties arise when multiple coin flips are under discussion—as for example, in the OP's question, which involves a situation where, in four flips of a coin, the coin came up "heads" once and "tails" three times.
The problem of how to properly express "heads" and "tails" in singular form and in plural form is especially evident in this case because it involves setting up "one heads" in contrast to "three tails"; but given the underlying reality that a single flip of a coin yields "heads" once or "tails" once, we would have no less difficulty explaining why—since we sometimes refer to "one heads"" or "one tails" as the result of a single flip—we don't refer to getting two results of "heads" and two results of "tails" in four flips as "two headses and two tailses." After all, the normal-sounding wording "two heads and two tails" doesn't add a plural ending to the singular ending "heads" or "tails" that we use to describe the result of a single flip.
The answer here, I believe, is that the description of multiple coin flip results isn't a progression from singular to plural "heads" or "tails" at all. The singular/plural difference in the underlying idea isn't "one head"/"two or more heads" and "one tail"/"two or more tails," but rather (in subsumed form)"one [result of] heads"/"two or more [results of] heads" and "one [result of] tails"/"two or more [results of] tails."
Basically our beef is properly with the wording "one tails" or "two tails" itself: Because we drop the words "result of" or "results of" from the complete way of expressing the phrase, we may look at a description of the results of multiple flips and suddenly think that we're presented with a singular/plural problem. But really we just have a generally accepted short form of an idea that makes perfect sense.
Still, if it bothers you to say
the probability of flipping exactly one heads and three tails
you can avoid the seeming number/plural inconsistency by rewording the phrase along the lines of
the probability of getting exactly one result of heads and three results of tails in four flips