We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{α_1, α_2, \cdots, α_m\}$, while the letters of the other word are i.i.d. drawn from $\{α_1, α_2, \cdots, α_m, α_{m+1}\}$, with probability $p > 0$ to be $α_{m+1}$, and $(1-p)/m > 0$ for all the other letters. The order of the variance of this length is shown to be linear in $n$.