binomial distribution

statistics video

This uses n choose k formula plus probabilities to determine the likelihood that something happens.

$P(k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)\cdot p^k\cdot (1-p{)}^{n-k}$

In english: The probability of exactly k successes is the number of arrangements (${}_n{C}_k$) times the probability of success ($p$) to the power of the number of successes times the probability of failure ($1-p$) to the power of the number of failures.

Binomial random variables

To be a binomial random variable, it must qualify against these 5 criteria:

1. The experiment consists of $n$ identical trials.
2. There are only two possible outcomes on each trial (S=success, F=failure)
3. The probability of S remains the same from trial to trial. This probability of this is $p$ and F is $q=1-p$
4. The trials are independent
5. The binomial random variable x is the number of S’s in n trials.

For binomial random variables, these are the relevant calculations:

$$\begin{gathered} \mu =np \\ \sigma =\sqrt{npq} \end{gathered}$$

where $q=1-p$.