= Solution
The two colonies with no observed cheaters are <right-censored> after seven days. Each contributes $\Pr(X>7)=(1-p)^7$, not the <probability> of an event on day seven or eight. The four observed first events contribute four factors of $p$ and $(6-1)+(7-1)+(5-1)+(5-1)=19$ failures. Including the two censored exposures gives
$$
\mathcal L(p)\propto p^4(1-p)^{19+7+7}=p^4(1-p)^{33}.
$$
The <beta distribution> prior therefore yields
$$
\boxed{p\mid D\sim\operatorname{Beta}(9,38)},\qquad
\mathbb E[p\mid D]=\frac9{47}\simeq0.19149.
$$
Its <probability density function> is proportional to $p^8(1-p)^{37}$ for $0<p<1$: it vanishes at both endpoints, is unimodal, and has its <mode> at $8/45\simeq0.17778$. The right panel of the sketch in question 5(b) shows this moderately right-skewed density. The calculation assumes daily observations detect the first event and that stopping the experiment after seven days is <independent censoring>.
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