# 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}\]

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*