Finding Binomial Probabilities

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