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 returns are selected at random and examined, find the probability that $6,7,$ or 8 of them contain an error.

Key Concepts

Probability Mass Function (PMF)

The probability mass function of the binomial distribution gives the probability that the random variable equals a specific value. 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 formula is fundamental for determining the likelihood of any particular outcome in binomial experiments.

Cumulative Probability for Discrete Outcomes

When calculating the probability of a range of outcomes (for example, the probability that exactly 6, 7, or 8 successes occur), it is necessary to compute the individual probabilities for each value using the PMF and then sum these probabilities. This approach allows one to determine the overall likelihood of a set of specific outcomes occurring in a binomial experiment.

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is relevant to situations where the outcome can be classified in a binary manner (for example, error or no error), and it provides the probability of achieving a given number of successes over many trials.

Independent Events

In the context of the binomial distribution, the assumption of independent events means that the result of one trial does not affect the outcome of any other trial. This independence is critical for applying the binomial model, as it ensures that the probability of success remains constant across all trials.

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