Editing Maximum likelihood estimation (section)

=== Discrete distribution, finite parameter space ===
Suppose one wishes to determine just how biased an [[unfair coin]] is.  Call the probability of tossing a '[[Obverse and reverse|head]]' ''p''. The goal then becomes to determine ''p''.

Suppose the coin is tossed 80 times: i.e. the sample might be something like ''x''<sub>1</sub>&nbsp;=&nbsp;H, ''x''<sub>2</sub>&nbsp;=&nbsp;T, ..., ''x''<sub>80</sub>&nbsp;=&nbsp;T, and the count of the number of [[Obverse and reverse|heads]] "H" is observed.

The probability of tossing [[Obverse and reverse|tails]] is 1&nbsp;−&nbsp;''p'' (so here ''p'' is ''θ'' above). Suppose the outcome is 49&nbsp;heads and 31&nbsp;[[Obverse and reverse|tails]], and suppose the coin was taken from a box containing three coins: one which gives heads with probability ''p''&nbsp;=&nbsp;{{frac|1|3}}, one which gives heads with probability ''p''&nbsp;=&nbsp;{{frac|1|2}} and another which gives heads with probability ''p''&nbsp;=&nbsp;{{frac|2|3}}. The coins have lost their labels, so which one it was is unknown. Using maximum likelihood estimation, the coin that has the largest likelihood can be found, given the data that were observed. By using the [[probability mass function]] of the [[binomial distribution]] with sample size equal to 80, number successes equal to 49 but for different values of ''p'' (the "probability of success"), the likelihood function (defined below) takes one of three values:

<math display="block">\begin{align}
\operatorname{\mathbb P}\bigl[\;\mathrm{H} = 49 \mid p=\tfrac{1}{3}\;\bigr] & = \binom{80}{49}(\tfrac{1}{3})^{49}(1-\tfrac{1}{3})^{31} \approx 0.000, \\[6pt]
\operatorname{\mathbb P}\bigl[\;\mathrm{H} = 49 \mid p=\tfrac{1}{2}\;\bigr] & = \binom{80}{49}(\tfrac{1}{2})^{49}(1-\tfrac{1}{2})^{31} \approx 0.012, \\[6pt]
\operatorname{\mathbb P}\bigl[\;\mathrm{H} = 49 \mid p=\tfrac{2}{3}\;\bigr] & = \binom{80}{49}(\tfrac{2}{3})^{49}(1-\tfrac{2}{3})^{31} \approx 0.054~.
\end{align}</math>

The likelihood is maximized when {{mvar|p}}&nbsp;=&nbsp;{{frac|2|3}}, and so this is the ''maximum likelihood estimate'' for&nbsp;{{mvar|p}}.