Question

Under what conditions binomial tends of normal distribution

Answer 1

For the normal distribution to be a good fit, you need to have large nn, how large depends on the value of pp, because this relies on the central limit theorem. A binomial distribution is a sum of nn independent Bernoulli random variables with probability of pp. For very high or very low pp, Bernoulli is a very skewed distribution. When a distribution is highly skewed, you need larger values of nn if you are depending on the central limit theorem. There are some rules of thumb for how big nn needs to be, but they are just rules of thumb, not hard rules.