Binomial to Poisson Approximation 
So, a little context. The image you see is from the GCE A-LEVEL syllabus where they have defined the conditions for approximating binomial to poisson.
But why did they have mention that the approximation is only possible when n > 50 and np/mean < 5? What is special about these two conditions? Is there a graphical or a proof-based reason?
THANK YOU FOR THE ANSWER(S)
 A: That should be $np\geq 5$ and $n(1-p) \geq 5$ for any approximation to be taken on Binomial Distribution. The approximation can be attributed to Central Limit Theorem.
This type of approximation is defended in another way also. Binomial distribution has PMF:
$$ f(X = x) = nCx~ p^x (1-p)^{n-x} $$ with $x = 0, 1, \ldots, n$.
Now when $n \rightarrow \infty$ and $p \rightarrow 0$ the computing the quantity $nCx$ becomes cumbersome and $p^x$ and $(1-p)^{n-x}$ also become unnecessarily cumbersome. Then assuming $np = \lambda$ it becomes the mean of Poisson Distribution and the quantity $np(1-p) \rightarrow np$ as $1-p \rightarrow 1$ i.e. variance of the Binomial Distribution becomes the Variance of Poisson Distribution.
This is how the Binomial Distribution is approximated as Poisson Distribution. Binomial Distribution can be approximated as a Normal Distribution too.
A: Going by Prof G E P Box's quote "in statistics no model is perfect but some are useful", modeling data using Probability Distributions also fits this quote very well, one can use any distribution anywhere if the constraints based on assumptions are relaxed - but the estimation error will be varying depending upon what approximation is being used. These numbers like $50$, $5$ they are decided based on simulated runs on real data. Earlier not even 10 years back, it used to be $30$ as the number of samples which was used for Normal distribution to be assumed for any distribution. Generally speaking Statistics always prefer "More the Better". But these are all sample based Statistical methodology if we enter into Big Data then almost all the concepts of Statistics starting from Hypothesis Testing, Confidence Interval Estimation etc severely fail.
