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Magazine Chapter 4: Problem 102
A corporate Web site contains errors on 50 of 1000 pages. If 100 pages are sampled randomly without replacement, approximate the probability that at least one of the pages in error is in the sample.
Short Answer
Expert verified
The probability is approximately 0.9941.
Step by step solution
01
Define the Problem
We need to find the probability that at least one of the sampled pages is erroneous, given that there are 50 erroneous pages out of 1000 and we sample 100 pages without replacement.
02
Calculate Probability of No Error in Sample
To find the probability of at least one erroneous page, we first find the probability that no pages in the sample are erroneous. This is the probability that all 100 pages selected are from the 950 pages without errors. This is given by the hypergeometric distribution:\[P(\text{0 errors}) = \frac{\binom{950}{100}}{\binom{1000}{100}}\]
03
Approximate with Complement
Since calculating large binomials directly is computationally complex, we use the rule of complements. The probability of at least one error page is:\[P(\text{at least 1 error}) = 1 - P(\text{0 errors})\]
04
Use Approximation for Large Samples
For large population sizes and samples, the probability can be approximated using binomial probabilities due to computational complexity. We approximate using:\[P(\text{at least 1 error}) = 1 - \left( \frac{950}{1000} \right)^{100}\]
05
Calculate Result
Calculate the expression for the approximation:\[P(\text{at least 1 error}) = 1 - \left(0.95\right)^{100} \approx 0.9941\]
06
Interpret the Result
The probability of selecting at least one page with an error out of 100 sampled pages is approximately 0.9941.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Approximation
When dealing with probabilities of complex or large data sets, approximations can help simplify calculations. In the context of the web page sampling problem, directly calculating the exact probability using the hypergeometric distribution can be cumbersome due to the involvement of large-binomial coefficients. To overcome this, we approximate the probability using simpler forms, such as the binomial distribution.
Approximations work by simplifying assumptions. When the sample size is small compared to the total population, probabilities like \(\left(\frac{950}{1000}\right)^{100}\) become easier to compute. This approximation gives a value close to the true probability while drastically reducing computational complexity.
In many real-world problems, exact calculations are either inefficient or unnecessary, and approximation offers a practical alternative to gauge probabilities within acceptable margins of error.
Approximations work by simplifying assumptions. When the sample size is small compared to the total population, probabilities like \(\left(\frac{950}{1000}\right)^{100}\) become easier to compute. This approximation gives a value close to the true probability while drastically reducing computational complexity.
In many real-world problems, exact calculations are either inefficient or unnecessary, and approximation offers a practical alternative to gauge probabilities within acceptable margins of error.
Binomial Distribution
The binomial distribution is a common probability distribution used to model scenarios with two possible outcomes: "success" or "failure." Unlike the hypergeometric distribution, the binomial distribution assumes that each trial in the experiment is independent, which makes it simpler to use in many cases.
In the web page problem, each page can be considered as having a 'success' of being error-free or a 'failure' of having an error. The probability of selecting a page without an error is \(\frac{950}{1000}\), as there are 950 non-error pages out of 1000. When drawing 100 pages, the compounded probability for all to be error-free is modeled by a binomial-like calculation: \(\left(0.95\right)^{100}\).
In instances where the sample size does not significantly deplete the initial conditions (a concept known as sampling with replacement), it's feasible to use a binomial approach for calculations. This approach simplifies many real-world statistical problems, particularly when the sample size is a small fraction of the population.
In the web page problem, each page can be considered as having a 'success' of being error-free or a 'failure' of having an error. The probability of selecting a page without an error is \(\frac{950}{1000}\), as there are 950 non-error pages out of 1000. When drawing 100 pages, the compounded probability for all to be error-free is modeled by a binomial-like calculation: \(\left(0.95\right)^{100}\).
In instances where the sample size does not significantly deplete the initial conditions (a concept known as sampling with replacement), it's feasible to use a binomial approach for calculations. This approach simplifies many real-world statistical problems, particularly when the sample size is a small fraction of the population.
Complement Rule
The complement rule in probability is a powerful tool that allows us to calculate the probability of an event by subtracting the probability of its complement from one. This is especially useful when calculating the direct probability of an event is difficult or complex.
The complement of an event is all outcomes that are not part of the event itself. Hence, for calculating the probability of having at least one error in our sampled pages, it's easier to first find the probability that no pages contain an error and then subtract from one: \[P(\text{at least 1 error}) = 1 - P(\text{0 errors})\].
This rule is universally applicable and highlights the connection between an event and its complement, providing a simpler roadmap to solve probabilistic problems without getting bogged down by direct calculations when they seem too intricate.
The complement of an event is all outcomes that are not part of the event itself. Hence, for calculating the probability of having at least one error in our sampled pages, it's easier to first find the probability that no pages contain an error and then subtract from one: \[P(\text{at least 1 error}) = 1 - P(\text{0 errors})\].
This rule is universally applicable and highlights the connection between an event and its complement, providing a simpler roadmap to solve probabilistic problems without getting bogged down by direct calculations when they seem too intricate.
