Ignoring factors that do not depend on the parameters, the product of the two <binomial likelihoods> is
$$
L(\phi_0,\phi_1)
=\phi_0^{120}(1-\phi_0)^{80}
 \phi_1^{110}(1-\phi_1)^{110}.
$$
Including the sampling constants multiplies this by ${200\choose120}{220\choose110}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 2 / 1 / Solution

If $x$ successes occur in $n$ independent Bernoulli trials, the binomial likelihood for the success probability $p$ is proportional to $p^x(1-p)^{n-x}$.


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