I see, in this case I would try to model directly on the observed (i.e., 34), and since this number is the output of the linear function (observed = total population * unknown proportion $p$ * time of the sampling period), you have:
total population = 1 million, this is an estimation so you can also represent it as a prior distribution
unknown proportion, 0.0405 in your example but unknown, you can model it as a Beta distribution
time of the sampling period = a known value. For example in a 10 hours event you sample for 5 mins then this value is 5/(10*60) = 0.00833333333
and you can model your observed as a Poisson: observed ~ Poisson(total population * unknown proportion $p$ * time of the sampling period).
Hope this makes sense 