# In math/stats terminology, $1/p$ is the reciprocal of $p$. So what is $1-p$ called, if anything?

I just want to describe in plain text a function that has many $1-p$ terms: $\frac{(1-p_1) + p_2}{2} × (1-p_3)$.

• "The complement", or more accurately "the probability of the complementary event". – Glen_b Jun 5 '13 at 22:35

Assuming $p$ is the probability of an event, $1 - p$ is the probability of its complement.

If $p$ is not the probability of an event then I doubt that $1 - p$ has any special meaning or name.

• I agree. Also, I'd add that if p is the probability of an event, then it's also common to denote 1-p as q. That might help simplify describing the function. – Ellie Jun 5 '13 at 20:49
• I’ve come across a few other settings in which $1-p$ was called the complement of $p$; notably in arithmetic modulo 2 (and more generally, linear algebra over $\mathbb{Z}/(2)$, but also I think some other settings, though I can’t remember them now. – PLL Jun 5 '13 at 23:02

In addition to $1-p$ being the complement where $p$ is a probability.

There is also the general process of taking $1 - x$ or $0 - x$ or $c - x$ where $x$ is a variable and $c$ is a constant. This is sometimes referred to as reversing or reflecting a variable.

• The same word is used in an analagous way in geometry. The sum of a probability and its complement comes to 1; the sum of an angle and its complement comes to a right angle. So in probability, the complement of 0.8 is 0.2, and in geometry the complement of 75 degrees is 15 degrees. "Complement" effectively means "what extra is needed to make it complete"! – Silverfish Nov 19 '14 at 15:18

In addition to the answers mentioning that $1-p$ may be regarded as the complement of $p$, it might be useful to note that the reason $1/p$ has a special name is because $1$ is the multiplicative identity (so that $p \times \frac{1}{p} = 1$).

When it comes to addition, the identity is $0$, and the value with a similar special name to the reciprocal is the negation $-p$ (so that $p + (-p) = 0$). For this reason, there's no necessity for $1-p$ to have a special name in the general case.

• I don't see how $1-p$ not being an inverse makes it any less special. It still has some nice properties. For instance, $f(p) := 1-p$ implies $f(f(p)) = 1 - (1-p) = p$. If anything, taking a complement is made more similar to negating or inverting by having this property, so arguments for giving them a special name would also apply. – Joshua Shane Liberman Jun 8 '13 at 4:08
• @JoshuaShaneLiberman That is true, but it's also true for any other number: $f(p) := k-p$ implies $f(f(p)) = k-(k-p)=p$. It's not practical to have a special name for every possible value of $k$. Maybe a name for the general $k-p$ case might exist, but there's no reason for the specific $1-p$ case to have one. – Philip C Jun 8 '13 at 9:16