$5\%$ of the inhabitants of a city are cricket fans. Determine approximately the probability that a sample of $100$ inhabitants contains at least $8$ cricket fans?
In the solution it was given that :
Let $X$ be the number of cricket fans. Then $X\sim P(\lambda)$ with $\lambda=8$. So required probability is $P(X\ge 8)=1-P(X\le 7)$.
I am not getting why it is a Poisson distribution; if it is a Poisson distribution then how $\lambda=8$ ?
A Poisson distribution is an approximation that is valid in the limit that each event is low probability and you have a lot of events. In this case the event is that a person is a cricket fan, which has a probability of $0.05$ and you have $100$ events. The probability is not too low and the number of events not too high, so the distribution may not be very accurate, but it is not bad. The application of the Poisson distribution is incorrect. The expected number of cricket fans is $5$, so you should ask what is the chance that a Poisson variable with $\lambda=5$ has value $8$ or above.