# ...the probability of flipping exactly one head[s] and three tails

I've encountered the following math question:

Four fair coins are tossed at the same time. What is the possibility that the four coins will come up with only one head and three tails.

Besides wanting to change "possibility" to "probability," and "only" to "exactly," I wonder about the word head. In coin-flipping, we typically use this word in the plural (and I understand there's a post about the history of "heads or tails"). But always?

My gut tells me to keep it plural. However, a bit of digging turned up examples for both singular and plural used in this type of context.

For example, Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences contains the following:

For one head and three tails, the probability is the same as one tail and three heads.

It still sounds awkward to me, as do other usages in the same book, such as:

The probability of the first coin giving a head is ½.

I'd definitely avoid that phrasing, particularly for the college crowd.

• Have a look at english.stackexchange.com/a/53395/71783 Feb 7, 2015 at 16:43
• @Frank: yes, useful info about the general expression, but this is about describing the number of instances of heads and tails. Feb 7, 2015 at 16:45
• "One heads and three tails" is ungrammatical when the reference is to how four coins fall. But I suppose if you were referring to repeated tosses of a single coin, you could use "one heads" to mean that "Heads!" was called out one time. Feb 7, 2015 at 16:47
• Consider that even in a single flip of a coin we'd always say, jocularly, Heads I win, tails you lose. Heads is not a plural, neither is tails. They are just the words for what side up a coin falls during the flipping of a coin. Go with your gut. Feb 7, 2015 at 16:58
• Not much of an answer, but I agree completely with @Frank. Heads and tails are not plural nouns here, they’re invariable, uninflectionable words. I wouldn’t singularise them when used attributively, either: if you toss two coins, one landing on heads and the other on tails, you’d still have one heads coin and one tails coin, not one *head coin and one *tail coin. Feb 7, 2015 at 23:01

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

"Heads" and "tails" are adverbs/adjectives with idiomatic plurals, so there's no strictly grammatical way to convert them into singular nouns. Of course, the meaning of "one head and three tails" is obvious, but in most cases it would sound better to use the adjective form ("one comes up heads and three come up tails").

But in the context of a math problem, I think you could grant the writers some leeway and assume they are treating the terms "head" and "tail" as sort of extra-grammatical formal symbols - i.e., easily-recognizable variable names signifying the possible states of the coin ("1 head and 3 tails" as opposed to something abstruse like "1 coin in state A and 3 coins in state B"). In situations like this conciseness and clarity are generally more important than maintaining a conversational tone.