The Equation

The probability of getting k successes in n trials is:
\[\begin{aligned}
p(k) &= {n \choose k} p^k (1-p)^{n-k}
\end{aligned}\]

The Variables

Variable	Meaning
$n$	total number of trials
$k$	number of successful trials
$p(k)$	probability of $k$ successes
$n-k$	the number of failures
$p$	probability of success for a single trial
$1-p$	probability of failure for a single trial
${n \choose k}$	number of permutations with $k$ successes

Calculating $n$ choose $k$

\[\begin{aligned}
{n \choose k} &= \frac{n!}{k!(n - k)!}
\end{aligned}\]

Inequalities

To find the probability of less than $y$ successes, sum the probabilities from $k=0$ to $k=y-1$:
\[\begin{aligned}
p(k < y) &= \sum_{i=1}^{y-1} p(k_i)
\end{aligned}\]
To find the probability of getting greater than $y$ successes, sum the probabilities from $k=y+1$ to $k=n$:
\[\begin{aligned}
p(k > y) &= \sum_{i=y+1}^{n} p(k_i)
\end{aligned}\]
To find the probability of getting $y$ or fewer successes, sum the probabilities from $k=0$ to $k=y$:
\[\begin{aligned}
p(k \leq y) &= \sum_{i=1}^{y} p(k_i)
\end{aligned}\]
To find the probability of getting $y$ or more successes, sum the probabilities from $k=y$ to $k=n$:
\[\begin{aligned}
p(k \geq y) &= \sum_{i=y}^{n} p(k_i)
\end{aligned}\]
To find the probability of from $y$ to $z$ successes, sum the probabilities from $k=y$ to $k=z$:
\[\begin{aligned}
p(y \leq k \leq z) &= \sum_{i=y}^{z} p(k_i)
\end{aligned}\]

Mean and Standard Deviation

The mean and Standard Deviation of a Binomial Distribution are:
\[\begin{aligned}
\mu &= np\\
\sigma &= \sqrt{np(1-p)}
\end{aligned}\]
Source: Statistics For Dummies, 2nd edition