Blue Cross Health Insurance reported that 4.5 % of claims...

Blue Cross Health Insurance reported that $4.5 \%$ of claims forms submitted for payment after a complex surgical procedure contain errors. If 100 of these forms are chosen at random, what is the probability that fewer than 3 of them contain errors? Use the Poisson approximation to the binomial distribution.

Key Concepts

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is fundamental in probability when modeling scenarios where each trial can result in one of two outcomes (commonly termed success or failure) and it allows the calculation of probabilities for various counts of successes out of a given number of trials.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided the events occur with a known constant mean rate and independently of the time since the last event. It is frequently used when events are rare relative to the size of the interval or set of trials.

Poisson Approximation to the Binomial Distribution

Under conditions where the number of trials is large and the probability of success in any given trial is small, the Poisson distribution serves as a good approximation to the binomial distribution. The approximation simplifies calculations by replacing the binomial parameters with a single parameter, ? (lambda), which is the product of the number of trials and the probability of success, thereby making it easier to compute probabilities for rare events.

Probability Mass Function (PMF) Computation

Both the binomial and Poisson distributions use a probability mass function (PMF) to assign probabilities to each possible outcome. In the case of the Poisson distribution, the PMF is given by P(X=k) = (e^(-?) * ?^k) / k!, where k is the number of occurrences. This formula is central to calculating cumulative probabilities, such as determining the probability of observing fewer than a specific number of events.

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