About one percent of all teller transactions at Chaste...

Name: Problem 7, Chapter 3: Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Uploaded: 2025-11-05T17:41:43-08:00
Duration: 1 min 1 s
Channel: Hoan Nguyen
Description: About one percent of all teller transactions at Chaste National Bank have a certain type of error. How large a random sample (with replacement) must be taken if the probability of its containing at least one transaction with an error is to be not less than 0.95? [Hint: Use the Poisson distribution.]

About one percent of all teller transactions at Chaste National Bank have a certain type of error. How large a random sample (with replacement) must be taken if the probability of its containing at least one transaction with an error is to be not less than 0.95? [Hint: Use the Poisson distribution.]

Key Concepts

Sample Size Calculation for Probabilistic Guarantees

Determining the minimum sample size needed to achieve a specified probability of observing at least one event involves setting up and solving an inequality based on the desired confidence level and the probability of the event in a single trial. This calculation is central to planning experiments or quality control processes where a high probability of detecting rare events is required.

Sampling with Replacement

Sampling with replacement refers to the process where each sample is returned to the population before the next draw, ensuring that each draw is independent and has the same probability distribution. This concept is critical when applying probabilistic models such as the Poisson distribution in scenarios where the occurrence of events is assumed to be independent from trial to trial.

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, under the assumption that these events occur with a constant mean rate and independently of the time since the last event. It is especially useful for modeling rare events, where the number of occurrences is low relative to the number of opportunities.

Complement Rule for 'At Least One' Events

When calculating the probability of at least one occurrence of an event, it is often simpler to compute the probability of the complementary event—namely, the probability that no events occur—and then subtract this value from 1. This technique greatly simplifies calculations in many probabilistic scenarios, including those modeled by the Poisson distribution.

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