Final answer:
The shape of the binomial distribution approaches that of a normal distribution as the sample size increases, and normal approximation is appropriate when np and nq are both greater than five. For large n and small p, the Poisson distribution can approximate the binomial distribution.
Step-by-step explanation:
As the sample size increases in a binomial distribution for a given value of probability of success p, the distribution's shape changes. Provided the conditions for a binomial distribution are met, which include a set number of independent trials, dichotomous outcomes (success or failure), and a constant probability of success p across trials, the binomial distribution tends to appear more similar to the normal distribution as the sample size increases. However, for the normal approximation to be suitable, the products of np and nq (with q being the probability of failure, or 1 - p) both need to be greater than five, and the approximation gets better if they are at least ten.
When p is low and the number of trials n is large, a Poisson distribution may be used as an approximation to the binomial distribution. Specifically, the rule of thumb is that a Poisson approximation is appropriate when n ≥ 20 and p ≤ 0.05. Moreover, the shift in distribution shape due to changes in n can be illustrated using the central limit theorem, which predicts that as the sample increases, the distribution of sample means will tend toward a normal, bell-shaped curve.