When the number of trials (n) is large but the probability of success (p) is small, Binomial probability distribution can be approximated using:

Correct option is C

The Poisson Distribution is used to approximate the Binomial Distribution when the number of trials ( n) is large, the probability of success ( p) is small, and the product of n×pn \times pn×p is finite (denoted as λ, the mean of the Poisson distribution). This approximation works well because, under these conditions, the distribution of the number of successes in the binomial distribution tends to converge to a Poisson distribution.
This relationship is particularly useful in modeling rare events over a large number of trials.
Information Booster:
· Poisson Distribution:
· It is discrete and describes the probability of a given number of events occurring within a fixed interval.
· The key parameter is λ (lambda), which represents the average number of occurrences.
· When to Use:
· If n>20 and p<0.05, the binomial distribution can be approximated by a Poisson distribution.
Additional Knowledge:
· Normal Distribution (Option a): Approximates a binomial distribution when nnn is large, and ppp is neither too small nor too large (close to 0.5).
· Hypergeometric Distribution (Option b): Used when the sampling is done without replacement, and the population size is finite.
· Bernoulli Distribution (Option d): Describes a single trial with two possible outcomes (success or failure). It is not used for approximating the binomial distribution.