Uncertainty on a poisson distributed probability?

Question

I know that the standard deviation is the square root of the mean if the random variable is poisson distributed, but do you assign that to the probability of a certain event? Let's say that for the random variable X, the mean is 4.6 with r event = 3. I calculate the probability to be 0.163. Should I assign the square root of 4.6 as an uncertainty to this probability? My teacher is very strict with uncertainties so I feel...uncertain about reporting probabilities without one.