This paper examines how a Bayesian decision maker would update his/her probability $p$ for the occurrence of an event $A$ in the light of a number of expert opinions expressed as probabilities $q_1, \cdots, q_n$ of $A$. It is seen, among other things, that the linear opinion pool, $\lambda_0p + \sum^n_{i = 1} \lambda_iq_i$, corresponds to an application of Bayes' Theorem when the decision maker has specified only the mean of the marginal distribution for $(q_1, \cdots, q_n)$ and requires his/her formula for the posterior probability of $A$ to satisfy a certain consistency condition. A product formula similar to that of Bordley (1982) is also derived in the case where the experts are deemed to be conditionally independent given $A$ (and given its complement).