n-gram
\[\begin{aligned} P(\text{<BOS>, I, love, to, play, <EOS>})&=P(\text{<EOS>}|\text{<BOS>, I, love, to, play})P(\text{<BOS>, I, love, to, play}) \\ &=P(\text{<EOS>}|\text{<BOS>, I, love, to, play})P(\text{play}|\text{<BOS>, I, love, to})P(\text{<BOS>, I, love, to}) \\ &=P(\text{<EOS>}|\text{<BOS>, I, love, to, play})P(\text{play}|\text{<BOS>, I, love, to})P(\text{to}|\text{<BOS>, I, love})P(\text{<BOS>, I, love}) \\ &=P(\text{<EOS>}|\text{<BOS>, I, love, to, play})P(\text{play}|\text{<BOS>, I, love, to})P(\text{to}|\text{<BOS>, I, love})P(\text{love}|\text{<BOS>, I})P(\text{<BOS>, I}) \\ &=P(\text{<EOS>}|\text{<BOS>, I, love, to, play})P(\text{play}|\text{<BOS>, I, love, to})P(\text{to}|\text{<BOS>, I, love})P(\text{love}|\text{<BOS>, I})P(\text{I}|\text{<BOS>})P(\text{<BOS>}) \\ \end{aligned}\] \[P(\text{<EOS>}|\text{<BOS>, I, love, to, play})\approx\frac{\text{COUNT}(\text{<BOS>, I, love, to, play, <EOS>})}{\text{COUNT}(\text{<BOS>, I, love, to, play})}\] \[P(x_n|x_{<n})\approx\frac{\text{COUNT}(x_1,\cdots,x_n)}{\text{COUNT}(x_1,\cdots,x_{n-1})}\]Markov Assumption
\[\begin{aligned} P(x_n|x_{<n})&\approx{P(x_n|x_{n-1},\cdots,x_{n-k})} \\ &\approx\frac{\text{COUNT}(x_{n-k},\cdots,x_n)}{\text{COUNT}(x_{n-k},\cdots,x_{n-1})} \end{aligned}\]if $k=2$ ,
\[\begin{aligned} P(x_n|x_{<n})&\approx{P(x_n|x_{n-1},x_{n-2})} \\ &\approx\frac{\text{COUNT}(x_{n-2},x_{n-1},x_n)}{\text{COUNT}(x_{n-2},x_{n-1})} \end{aligned}\]