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Independent Jeffreys priors and the two independent binomial distribution likelihood factors give, by Beta-binomial conjugacy,The Bayesian posteriors remain independent because each observation factor involves only its own probability. Their posterior means are respectively $0.7$ and $0.3$. Notice that $p_T$ is the probability of saying milk first when tea was first, rather than the probability of a correct tea identification.

For the plug-in Neyman allocation, the estimated Bernoulli distribution variances are $6/25$ and $3/16$, soThis uses the target ratio directly as the allocation odds; a procedure that actively corrects previous allocation imbalances would need an additional rule.

For the randomized play-the-winner rule RPW$(1,1)$, return the drawn ball and add one ball of the same treatment after a success or of the opposite treatment after a failure. Treatment 0 has three successes and two failures, and treatment 1 has one success and three failures. The urn therefore has $1+3+3=7$ treatment-0 balls and $1+1+2=4$ treatment-1 balls. HenceFor dynamic programming with only the final patient left, there is no future value from learning. Independent uniform prior distributions and Beta-binomial conjugacy give posteriors $p_0\mid \mathcal{D}\sim \mathrm{Beta}(4,3)$ and $p_1\mid \mathcal{D}\sim \mathrm{Beta}(2,4)$. Their posterior predictive probabilities of success are $4/7$ and $2/6=1/3$. The optimal terminal action therefore givesThis is optimal for expected successes with no imposed lower bound on the randomization probabilities.

Let $S={\sum }_{i=1}^n{X}_i$ be the number of heads. The likelihood function for the Bernoulli distribution sample is proportional to ${\theta }^S(1-\theta {)}^{n-S}$. A uniform distribution prior on $[0,1]$ is the $\mathrm{Beta}(1,1)$ distribution, so Beta-binomial conjugacy gives the posterior distribution

The uniform prior is the Beta distribution $\mathrm{Beta}(1,1)$. By Beta-binomial conjugacy, after observing $X$ heads the posterior isThe Bayes estimator under squared error loss is the posterior mean, hence

LetThus $d_t$ patients recover during month $t$, while $r_t-d_t$ survive that month without recovery. The likelihood function is

The independent $\mathrm{Beta}(T,1)$ prior densities are proportional to $q_t^{T-1}$. Beta-binomial conjugacy therefore leaves the coordinates posteriorly independent, with

A posterior predictive probability averages a future-event probability over the Bayesian posterior. For a Bernoulli distribution with unknown parameter $p$, it is $\mathbb{E}[p\mid \mathcal{D}]$. Under Beta-binomial conjugacy with posterior $\mathrm{Beta}(a,b)$, the predictive success probability is $a/(a+b)$.