Suppose that, on average, 1 person in 1000 makes a numerical...

Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 forms are selected at random and examined, find the probability that $6,7,$ or 8 of the forms contain an error.

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Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. This concept is essential when determining probabilities of a specific number of occurrences, such as errors on tax forms, where each case is an independent trial with a small but constant chance of error.

Probability Mass Function

The probability mass function (PMF) for the binomial distribution provides the probability of obtaining exactly a specified number of successes. It is calculated using the formula P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k)), where n is the number of trials, k is the number of successes, and p is the probability of success. This function is key for computing the probability of outcomes in discrete scenarios.

Summation of Probabilities

When calculating the probability of a range of outcomes, such as having 6, 7, or 8 errors, it is necessary to compute the individual probabilities for each outcome using the PMF and then sum these probabilities. This method leverages the additive property of probabilities in mutually exclusive events, making it a fundamental concept in probability theory.

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