Suppose (x) is a random variable with at least two levels, with (operatorname{Pr}left(x=x_{i} ight)=p_{i}), for (i=1,2). Let

Suppose \(x\) is a random variable with at least two levels, with \(\operatorname{Pr}\left(x=x_{i}\right)=p_{i}\), for \(i=1,2\). Let \(x^{\prime}\) be the new random variable based on \(x\) with the two levels \(x_{1}\) and \(x_{2}\) combined, i.e.,

\[x^{\prime}=\left\{\begin{array}{l}x_{1} \text { if } x=x_{1} \text { or } x_{2} \\x \text { if otherwise }\end{array}\right.\]

then \(H(x)=H\left(x^{\prime}\right)+\left(p_{1}+p_{2}\right) H(z)\), where \(H(z)\) is the entropy conditional on \(x=x_{1}\) or \(x_{2}\), or, in other words, \(z \sim \operatorname{Bernoulli}\left(\frac{p_{1}}{p_{1}+p_{2}}\right)\).