Binomial Distribution/Example/Arbitrary Example 1

Example of Binomial Distribution

Let a die be cast $4$ times.

Let a score of $6$ be denoted as a success.

Then the experiment can be modelled by a binomial distribution $\Binomial n p$ such that $n = 4$ and $p = \dfrac 1 6$.

Thus the probability of $2$ successes is $\dfrac {25} {216}$.

Let $\map P k$ be the probability of $k$ successes.

Then by definition of binomial distribution:

$\map P k = \dbinom 4 k \paren {\dfrac 1 6}^k \paren {\dfrac 5 6}^{4 - k}$

Hence:

$\ds \map P 2$

$=$

$\ds \dbinom 4 2 \paren {\dfrac 1 6}^2 \paren {\dfrac 5 6}^2$

$\ds $

$=$

$\ds 6 \times \dfrac 1 {6^2} \dfrac {5^2} {6^2}$

$\ds $

$=$

$\ds \dfrac {5^2} {6^3}$

$\ds $

$=$

$\ds \dfrac {25} {216}$

$\blacksquare$