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Chapter 13: Problem 59

If the DNA profile (or combination of alleles) found on the hair is possessed by one in \(1.6\) million individuals, and the database of convicted felons contains \(4.5\) million individuals, approximately how many individuals in the database would demonstrate a match bet ween their DNA and that found on the hair?

Short Answer

Expert verified
Approximately 3 individuals would demonstrate a match.

Step by step solution

01

Understand the problem

We need to find how many individuals in a database of convicted felons possess a specific DNA profile that is found in 1 out of 1.6 million people.
02

Calculate the probability of a match

The probability that any one person from the database has the DNA profile is given as 1 in 1.6 million. This probability can be expressed as \( \frac{1}{1,600,000} \).
03

Calculate the expected number of matches

To find the expected number of people with this DNA profile in the database, multiply the total number of individuals in the database by the probability: \( 4,500,000 \times \frac{1}{1,600,000} \).
04

Perform the multiplication

Calculate \( \frac{4,500,000}{1,600,000} \), which simplifies to \( 2.8125 \).
05

Interpret the result

The result \( 2.8125 \) means that, statistically, we expect approximately 2.81 individuals in the database to have a matching DNA profile, i.e., roughly 3 individuals.

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Key Concepts

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

Understanding Probability
Probability is a fundamental concept in statistics that measures the likelihood of an event happening. It's expressed as a number between 0 and 1, where 0 means the event cannot happen, and 1 means it will definitely happen. In many real-world situations, such as DNA profiling or casino games, probability helps us estimate outcomes.
  • A probability of 0.5 indicates there's an equal chance of either happening.
  • In our example, the probability of randomly picking one individual with the relevant DNA profile from the global population is \( \frac{1}{1,600,000} \).
This is a very small probability, indicating that the event is rare. Understanding this helps in appreciating how statistical probabilities can sometimes lead to unexpected results in large groups of data.
Calculating Expected Value
Expected value is a critical concept used to predict the average outcome when any random action is repeated many times. In simple terms, it's like asking, "If I did this repeatedly, what should I expect to happen on average?" This is incredibly useful in scenarios like financial investments or predicting the number of specific individuals in a database.In the context of DNA profiling, the expected value helps calculate how many people in a large database might match a specific DNA profile.
  • Given the probability found earlier and a database size, the expected value is calculated by multiplying the probability by the total number of subjects, i.e., \( 4,500,000 \times \frac{1}{1,600,000} = 2.8125 \).
  • The result, 2.8125, suggests that on average, about 3 people in that database can be expected to have this DNA profile.
This expectation doesn't guarantee exact results every time but provides a solid statistical estimate useful for planning and analysis.
The Role of DNA Profiling
DNA profiling, also known as DNA fingerprinting, is a powerful technique used in forensic science to identify individuals based on their DNA sequence. The method relies on analyzing specific regions of DNA that may vary greatly among people.
  • It's used in law enforcement for identifying suspects, cadavers, or in paternity tests.
  • It involves matching the DNA profile from a crime scene to profiles stored in a database.
In our example, a person with a rare DNA type "1 in 1.6 million" means that in any given sample of the population, their profile should appear infrequently. DNA profiling enables narrowing down suspects or identifying individuals, but it's crucial to interpret results cautiously due to probabilities and expected values involved.
Law Enforcement Databases
Law enforcement databases, such as those used globally, store DNA profiles of individuals related to criminal activities. This stronghold of data is pivotal for solving crimes because it helps match crime scene evidence to potential suspects.
  • These databases often include profiles from convicted criminals, missing persons, or unidentified remains.
  • In the step-by-step exercise, the database contains 4.5 million profiles.
The importance of such databases grows with their size and the comprehensiveness of the entries. However, privacy concerns can arise, and statistical prowess is needed to ensure the correct interpretation of DNA matches.

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