How to determine whether a problem belongs to the poisson distribution?

Question

I was solving my university past paper and came about this question:

A microbiologist wants to estimate the concentration of a certain type of bacteriumin a wastewater sample.
She puts a 0.5 mL sample of the waste-water on a microscope slide and counts 39 bacteria.
Estimate the concentration of bacteria per mL, in this waste-water.

Ater searching my textbook I figured out that this problem belongs to Poisson distribution.But my question is how are we supposed to identify to which distribution the problem belongs to? I know the general rule to identify questions that belong to Poisson distribution:They should have small success probability and large sample size.Also the mean would be equal to the variance.Any help would be helpful for my End sem exam.

Answer 1

First, your reasoning that "They should have small success probability and large sample size. Also the mean would be equal to the variance." is wrong, this isn't related to the Poisson distribution, you can have other distributions sharing these properties and you can have a Poisson distribution without these properties so I don't advise you to look for these as they are unrelated. Similarly, having small success probabilities and large sample size is unrelated.

Second, a Poisson distribution is when you are given on average how much an event occurs in a certain interval, and you want to find the probability that it occurs another number of times (different that than the mean) in the interval. The interval could be time, distance, weight, etc... anything. Moreover, the events should be independent.

Note that you are always given the mean explicitly or given enough information to find the mean yourself.

To illustrate this by example, assume you are given the following:

A call center receives 4 calls per hour, what is the probability it receives 6 calls per hour?

Here the events are independent since a customer calling doesn't affect the probability of another calling. You are given the mean (4 calls per hour) and you want to find the probability that a value other than the mean (6 calls) happens, so you use the Poisson distribution.

Note that they might not explicitly state the same intervals, so for example one variation of this problem could be the following:

A call center receives 5 calls per hour, what is the probability it receives 15 calls in the next 90 minutes?

Here you would next to convert what is given to match the mean, so 15 calls per 90 minutes is the same as 10 calls per an hour, then you work.

Going back to your example:

A microbiologist wants to estimate the concentration of a certain type of bacteriumin a wastewater sample. She puts a 0.5 mL sample of the waste-water on a microscope slide and counts 39 bacteria. Estimate the concentration of bacteria per mL, in this waste-water.

This problem can be solved with simple reasoning by just doing $\frac{39}{0.5}$ as if you have x amount of y/2 total you will have 2x amount in y (just multiplying both sides by 2, no distributions involved)