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I'm starting studying maximum likelihood estimation, and I have came across to a problem that I don't know how to solve yet.

How can I write the likelihood function to test the hypothesis that some treatment reduced the proportion of something bad to happen when compared to a control group? I understand I am dealing with two binomial distributions, but I can't think on how to write the likelihood.

If I had, for instance, this data:

\begin{tabular}{c|cc|c} & Control & Treatment & Total Bad & 51 & 12 & 63 Good & 595 & 611 & 1206 Total & 646 & 623 & 1269 \end{tabular}

$p_C = .0789 $ and $p_T = .0192$

How do I write a likelihood to compare these two numbers? If I wanted to find the MLE of $p_C$ and $p_T$ analytically how should I do it?

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You just write down your model! If the sampling are independent in the treatment and control groups, you just have a product of two independent binomial likelihood!

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