Binomial Distribution/Example/Arbitrary Example 1
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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 $\dfrac {25} {216}$.
Proof
The experiment of casting a die can be modelled as a Bernoulli process.
Hence from Bernoulli Process as Binomial Distribution, the model is appropriate.
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:
\(\ds \map P 2\) | \(=\) | \(\ds \dbinom 4 2 \paren {\dfrac 1 6}^2 \paren {\dfrac 5 6}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6 \times \dfrac 1 {6^2} \dfrac {5^2} {6^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5^2} {6^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {25} {216}\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial distribution