 Concentration of a beta prior near zero

For $\Theta \sim \mathrm{Beta}(1,\beta )$, direct integration of its probability density function gives the displayed tail for $0<t<1$. Increasing $\beta$ strictly decreases every such tail probability, and $\Theta$ converges in probability to zero as $\beta \to \infty$. Its mean is $1/(\beta +1)$ and its variance is $\beta /[(\beta +1{)}^2(\beta +2)]$; both decrease for $\beta >1$. For the variance, logarithmic differentiation gives $1/\beta -2/(\beta +1)-1/(\beta +2)<0$ in that range.

After $S$ successes in $N$ conditionally independent Bernoulli trials, Beta-binomial conjugacy gives posterior mean $(S+1)/(N+\beta +1)$. Its derivative with respect to $\beta$ is negative, while the weight $(\beta +1)/(N+\beta +1)$ assigned to the prior mean is increasing for $N>0$.