Final answer:
A binomial distribution represents the number of successes in a fixed number of independent trials, with only two possible outcomes: success or failure. The mean and standard deviation can be calculated using formulas. The Poisson distribution can be used to approximate the binomial if certain conditions are met.
Step-by-step explanation:
A binomial distribution is a probability distribution that represents the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. The probability of success on a single trial is denoted by p. The random variable X represents the number of successes in n trials. The mean of X is given by µ = np, and the standard deviation is given by σ = √(npq).
The Poisson distribution can be used to approximate the binomial distribution if the probability of success on a single trial is small and the number of trials is large. To ensure a good approximation, the rules of thumb state that np > 5 and nq > 5.
Technology tools like calculators and software can be used to easily calculate binomial probabilities, eliminating the need for normal approximation. It's important to remember the conditions for a binomial experiment (fixed number of trials, independent trials, same probability of success) and verify the shape of the distribution to ensure it is similar to the normal distribution.