BINOMIAL DISTRIBUTION

\(\displaystyle\Rightarrow\) It is a discrete probability distribution which gives the theoretical probabilities.

\(\displaystyle\Rightarrow\) Binomial distribution allow us to deal with circumstances in which the outcomes belong to two relevant categories such as success or failure.

\(\displaystyle\Rightarrow\) It depends on the parameter p or q i.e. the probability of success or failure and n (i.e. the number of trials). The parameter n is always a positive integer.

\(\displaystyle\Rightarrow\) The distribution will be symmetrical if \(\displaystyle{p}={q}\). It is skew-symmetric or asymmetric if \(\displaystyle{p}\ne{q}\).

\(\displaystyle\Rightarrow\) The possibility of outcome of any trial does not change and is independent of the results of previous trials.

Formula

\(\displaystyle{P}{\left({x}\right)}=^{{{n}}}{C}_{{{x}}}\cdot{p}^{{{x}}}\cdot{q}^{{{n}-{x}}}\) where \(\displaystyle{x}\Rightarrow\) number of successes

Step 2

HYPER GEOMMETRIC DISTRIBUTION

\(\displaystyle\Rightarrow\) A hypergeometric distribution has a specified number of dependent trials having two possible outcomes, success or failure.

\(\displaystyle\Rightarrow\) The random variable is the number of successful outcomes in the specified number of trials.

\(\displaystyle\Rightarrow\) In statistics and probability theory, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes in draws without replacement.

\(\displaystyle\Rightarrow\) The probability of a success is not the same on each trial without replacement, thus events are not independent.

Formula

\(\displaystyle{P}{\left({x}\right)}={\frac{{{s}{C}_{{{k}}}\cdot^{{{N}-{S}{C}_{{{N}-{S}}}}}}}{{{N}{C}_{{{n}}}}}}\)

where \(\displaystyle{S}\Rightarrow\) successes from population

\(\displaystyle{N}\Rightarrow\) population size

\(\displaystyle{n}\Rightarrow\) sample size

\(\displaystyle{k}\Rightarrow\) successes from sample