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Chapter 13: Problem 5

Lost Internet References. Internet sites often vanish or move so that references to them can't be followed. In fact, \(47 \%\) of Internet sites referenced in major medical journals are lost. 5 If a paper contains seven Internet references, what is the probability that all seven are still good? What specific assumptions did you make to calculate this probability?

Short Answer

Expert verified
The probability is approximately 0.0127. Assumed independence of references.

Step by step solution

01

Define the Probability of a Good Site

In this problem, it is given that 47% of Internet sites in medical journals are "lost" or no longer available. This means that 53% of sites are still good and can be accessed. Therefore, the probability that a single internet reference is still good is 0.53.
02

Calculate Probability for One Reference

The probability that one specific Internet reference is still good (accessible) is given by: \( p = 0.53 \).
03

Assume Independence of References

We assume that the accessibility of each Internet reference is independent of the others. This is critical because it allows us to multiply probabilities across different references.
04

Expression for All References Being Accessible

We express the probability that all seven references are still accessible using the independence assumption. The probability that all seven references are still good is \( p^7 \).
05

Perform Calculation

Substitute the known values into the expression: \( 0.53^7 \). Calculate this value to find the probability that all seven references are still good.
06

Calculate Probability

When we calculate the expression \( 0.53^7 \), we get approximately 0.0127. Therefore, the probability that all seven references are still good is about 0.0127.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independence Assumption
When dealing with probabilities, especially in cases where multiple events occur, the independence assumption is a key concept. It means that the likelihood of one event happening does not affect the occurrence of the other. This is crucial in our Internet reference problem.

For example, consider each Internet reference as a separate event with its own probability of being accessible. Assuming independence means each site's accessibility does not impact another's status.
  • This allows us to multiply the probabilities of individual events to ascertain the overall probability.
  • In our case, we assumed that each of the seven Internet references functions without being influenced by the outcomes of the others.
This assumption greatly simplifies the calculations, allowing us to just raise the probability of one site's accessibility to the power of the number of references, i.e., seven.
Internet References
Internet references are a common feature in academic papers and research articles. They direct readers to sources that support the information presented.

However, as time passes, websites can become inaccessible for a variety of reasons, leading to what are called 'lost references.' These lost URLs can impact the reliability and verifiability of a document.
  • The issue of disappearing or changing web addresses makes it relevant to consider such probabilities like in our exercise, highlighting the practical importance of maintaining or archiving content.
  • The exercise specifically quantifies the probability of such references being intact, hence providing insight into the reliability of internet-based citations in long-term research.
Binomial Probability
Binomial probability comes into play when we are dealing with experiments that involve repeated trials of a binary event, that is, one that has only two possible outcomes.
  • In our context, each Internet reference can either be good (accessible) or lost (not accessible).
  • The chance for a "success" or a "good" reference is given by a probability of 0.53, based on the fact derived from the given problem statement that 53% of sites are usually good.
This is essentially a binomial experiment with a fixed number of trials (7 references), and our task was to find the probability that all trials result in a success. Using the binomial probability model, we calculate the overall probability by raising the single event probability to the number of trials.
Step-by-Step Solution
The beauty of a step-by-step solution is in its systematic approach, breaking down complex problems into manageable parts.

In the exercise, we:
  • First, identified the probability of a single success event (in this case, finding a good reference) as 0.53.
  • Next, assumed each trial (each reference being either good or lost) was independent.
  • Then, expressed the cumulative probability using the formula for independent events: multiplying the probability of a single event occurring the required number of times (raising 0.53 to the power of 7).
  • Finally, computed the final probability, arriving at approximately 0.0127, which quantifies the likelihood of finding all seven references intact.
This logical progression not only aids understanding but also enables accurate calculations, embodying both practicality and clarity in statistical evaluations.

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