Bayes probability confidence I use Bayes theorem to estimate the impact of a sales person on customer's decision to buy a product. 
$ P(buy|salesperson) = \frac{P(salesperson|buy) P(buy)}{P(salesperson)} $
Naturally, some salespersons are more active than others, which is taken care of by the denominator. However, I'm still having hard time to interpret similar values of $P(buy|salesperson)$  for two representatives with vastly different values of $P(salesperson)$. Take for example the following situation:


*

*A dataset contains 1000 interactions between sales representatives and potential customers

*Salesperson A performed 200 interactions ($P(salesperson A) = 0.2$)

*Salesperson B performed 2 interactions ($P(salesperson B) = 0.002$)

*The posterior probability $P(buy|salesperson)$ of the two representatives is identical


Am I less confident in the posterior probability calculated for salesperson B (I assume I am)? How do I communicate this confidence? 
NOTE there are more salespersons that A and B and there are more buying events than those performed by A and B. At this point I'm only concerned about comparing the probabilities of these two ones
 A: Bayes theorem states that:
$$ P(A|B) = \frac{ P(B|A)\,P(A)}{ P(B|A) P(A) + P(B|\neg A) P(\neg A) } = 
\frac{ P(B|A)\,P(A)}{ \sum_A P(B|A) P(A) }
$$
so if in your case $buy$ stays the same, there is no way $P(buy|salesperson)$ could be the same for both cases because numerator changes while denominator does not change. The denominator is a total probability, that is constant (it makes the probabilities to sum to $1$). By total probability we mean in here sum over $B$ given all possible $A$'s, so in your case salespersons interactions given all possible buys.
A: Your data are 200 interactions for salesperson A and 2 interactions for salesperson B. You state "the posterior probability of the two representatives is identical", but this isn't a posterior probability. What you really mean is that the empirical sale proportion is identical. 
To introduce notation, let $y_A$ be the number of sales for salespersonA out of $n_A=200$ interactions and $y_B$ be the number of sales for salespersonB out of $n_B=2$ transactions. What you have told us is $y_A/n_A=y_B/n_B$. 
Now, let $\theta_A$ and $\theta_B$ be the true sale probability for salesperson A and B, respectively. And let's assume independent Jeffreys priors for $\theta_A$ and $\theta_B$, i.e. both independently have a Beta(1/2,1/2) prior. Now (using Bayes theorem) the posterior for salesperson A's true sales probability is Be(1/2+y_A, 1/2+n_A) and the posterior for salesperson B's true sales probability is Be(1/2+y_B, 1/2+n_B). So yes, you have much less uncertainty in salesperson A's true sales probability than you have in salesperson B's true sales probability. 
