The Peak of the Posterior After Two Batches

A seed lab is estimating $\theta$, the probability that a seed from a new hybrid line germinates.

From last season's trials the lab adopts a $\text{Beta}(3, 2)$ prior on $\theta$.

This season the seeds are tested in two batches, and the lab updates sequentially — the posterior after the first batch becomes the prior for the second:

• Batch 1: $6$ seeds are planted and $5$ of them germinate.
• Batch 2: $4$ seeds are planted, and three times as many germinate as fail to germinate.

The Beta distribution is conjugate to the Bernoulli/binomial likelihood, so the final posterior is again a Beta distribution.

The lab now needs a single point estimate and chooses the MAP estimate — the mode of the final posterior.

What is that MAP estimate of $\theta$?

Round your answer to 4 decimal places.