Sampling model / likelihood using Poisson distribution in Bayesian inference?

I'm trying to solve a problem/question from my professor. The problem/question is we want to know the proportion of the population in city A who is infected with COVID-19. For that, an examination was taken on 500 people. Of the 500 people, 10 were infected with COVID-19. The problem will be solved using Bayesian inference. My professor state that the sampling model/likelihood that must be used is Poisson distribution.

But I am very confused about the PDF (Probability Density Function) form of the Poisson distribution for this problem and how to determine its parameters as well as the description of the parameters. Because in my opinion there is no known information about the time or space for the problem, so I don't think Poisson distribution is the right sampling model/likelihood to choose.

Can anyone help me in determining the PDF form and the description of the parameters of the Poisson distribution?

Each person is either infected (with probability $$p$$) or not infected. Thus it follows Bernoulli distribution. Then the probability that the number of infected persons among the population of $$n$$ follows binomial distribution. When $$n$$ is large and $$p$$ is small, the binomial distribution approaches (or is well approximated by) Poisson distribution with $$\lambda = np$$.