 Solution

Use prior elicitation to make the qualitative judgments quantitative: ask whether $10\%$ represents a mean, median or mode, and what probability “unlikely above $15\%$” should mean, for example a $5\%$ upper-tail probability. A beta distribution on $(0,1)$ is a convenient family. If $0.1$ is taken as the mean, write its parameters as $a_q=0.1\kappa$, $b_q=0.9\kappa$, then choose $\kappa$ so that the desired tail probability above $0.15$ is obtained. Check its quantiles and prior predictive distribution for plausibility, and assess sensitivity to reasonable alternative elicited values. The information suggests an informative prior near $0.1$, but does not uniquely determine its parameters.