= Solution
Let $Z$ be the unobserved part of the fourth count associated with the component of <probability> $\theta/3$. Its complementary subcount has <probability> $7/12$. Conditional on the observed fourth count and a parameter value,
$$
\boxed{Z\mid x_4,\theta\sim\operatorname{Bin}\left(x_4,\frac{4\theta}{7+4\theta}\right).}
$$
The complete-data <log-likelihood> now has the simple form
$$
\ell_c(\theta)=(x_3+Z)\log\theta+(x_1+x_2)\log(1-\theta)
+\mathrm{constant}.
$$
The constant term can depend on the completed counts, but not on $\theta$. The split therefore converts the troublesome $\log(7+4\theta)$ term into a binomial-like maximization problem. Since $Z$ is unobserved, it should be averaged conditionally in an <EM algorithm>, not guessed or treated as observed.
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