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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 between their DNA and that found on the hair?

Short Answer

Expert verified
Approximately 2.81 individuals would match.

Step by step solution

01

Understanding the Problem

We have a database of 4.5 million convicted felons. Each individual has a certain DNA profile, and one specific DNA profile appears in one out of every 1.6 million individuals. Our task is to find out how many individuals in this database would have a DNA match.
02

Write Down the Given Data

We are given that the DNA profile appears in 1 out of 1.6 million individuals. The size of our database is 4.5 million individuals.
03

Calculate the Probability of a Match

To find the probability, divide the number of individuals who could have the DNA profile in a general population by the total number of people in the general population: \(\text{Probability of DNA profile} = \frac{1}{1.6 \text{ million}}\).
04

Estimate the Number of Matches

To find out how many individuals in the database could have this DNA profile, multiply the size of the database by the probability of the DNA profile: \[\text{Expected number of matches} = 4.5 \text{ million} \times \frac{1}{1.6 \text{ million}} = \frac{4.5}{1.6}\].
05

Simplify the Calculation

Compute \(\frac{4.5}{1.6}\) to find the approximate number of individuals with the DNA profile. Calculating this gives us approximately 2.8125.

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Key Concepts

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

DNA Matching
DNA matching involves comparing profiles derived from biological material like hair to other profiles, often stored in databases, to find matches. Each individual has a unique DNA profile composed of alleles inherited from their parents. These genetic markers can identify individuals with high precision.
When a forensic investigation finds a strand of hair at a crime scene, they can analyze the allelic combinations (DNA profile) and compare it against a database of known profiles from convicted individuals. If this profile appears in only 1 out of 1.6 million people, the uniqueness makes it significant for matches.
To determine possible matches in a database, we calculate how often such a profile will appear, considering database size and profile rarity. This straightforward comparison relies on mathematical probability principles and often requires rounding in practical applications due to decimal results.
Statistical Probability
Statistical probability helps predict how often a particular DNA profile might appear in a group. Here, it involves taking the specific probability of one profile鈥�1 in 1.6 million鈥攁nd using it to estimate how many times such a profile occurs in a database of 4.5 million individuals.
We use the formula:
  • Probability of DNA profile = \( \frac{1}{1.6 \text{ million}} \)
  • Expected number of matches = Database size \( \times \) probability
  • Calculate: \( 4.5 \text{ million} \times \frac{1}{1.6 \text{ million}} = \frac{4.5}{1.6} = 2.8125 \)
This calculation tells us that approximately 2.8125 individuals in the database might match that profile.
Probability provides a framework for predicting outcomes using known data. It's crucial for DNA matching due to the uniqueness of genetic profiles.
Database Analysis
Database analysis involves interpreting large sets of data to identify patterns and make predictions. When analyzing a database of convicted felons, each entry represents a unique DNA profile.
Understanding the size of the database鈥�4.5 million individuals鈥攊s crucial. The rarity of the DNA profile in a general population (1 in 1.6 million) helps estimate possible matches within the database.
By multiplying these factors, we efficiently analyze the database to assess the likelihood of matches. It's less about combing through each individual manually but using statistical principles to make educated predictions about data trends.
Combining technology and probability allows analysts to effectively utilize vast databases, streamlining investigations and maximizing their predictive capabilities.

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Most popular questions from this chapter

The probability of a flush. A poker player holds a flush when all five cards in the hand belong to the same suit (clubs, diamonds, hearts, or spades). We will find the probability of a flush when five cards are drawn in succession from the top of the deck. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well shuffled, each card drawn is equally likely to be any of those that remain in the deck. (a) Concentrate on spades. What is the probability that the first card drawn is a spade? What is the conditional probability that the second card drawn is a spade, given that the first is a spade? (Hint: How many cards remain? How many of these are spades?) (b) Continue to count the remaining cards to find the conditional probabilities of a spade for the third, the fourth, and the fifth card drawn, given in each case that all previous cards are spades. (c) The probability of drawing five spades in succession from the top of the deck is the product of the five probabilities you have found. Why? What is this probability? (d) The probability of drawing five hearts or five diamonds or five clubs is the same as the probability of drawing five spades. What is the probability that the five cards drawn all belong to the same suit?

Of people who died in the United States in recent years, \(86 \%\) were white, \(12 \%\) were black, and \(2 \%\) were Asian. (This ignores a small number of deaths among other races.) Diabetes caused \(2.8 \%\) of deaths among whites, \(4.4 \%\) among blacks, and \(3.5 \%\) among Asians. The probability that a randomly chosen death is a white person who died of diabetes is about (a) \(0.107\). (b) \(0.030\). (c) \(0.024\).

Screening job applicants. A company retains a psychologist to assess whether job applicants are sulted for assembly-line work. The psychologist classifies applicants as one of \(A\) (well suited), \(B\) (marginal), or \(C\) (not suited). The company is concerned about the event \(D\) that an employee leaves the company within a year of being hired. Data on all people hired in the past five years give these probabilities: \(P(A)=0.4 P(B)=0.3 P(C)=0.3 P(A\) and \(D)=0.1 P(B\) and \(D)=0.1 P(C\) and \(D)=0.2\) \(P(A)=0 . A \quad P(B)=0.3 \quad P(C)=0.3\) \(P(A\) and \(D)=0.1 \quad P(B\) and \(D)=0.1 \quad P(C\) and \(D)=0.2\) Sketch a Venn diagram of the events \(A, B, C\), and \(D\) and mark on your diagram the probabilities of all combinations of psychological assessment and leaving (or not) within a year. What is \(P(D)\), the probability that an employee leaves within a year?

Common Names. The U.S. Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez, and Wilson. These names account for \(9.6 \%\) of all U.S. residents. Out of curiosity, you look at the authors of the texthooks for your current courses. There are nine authors in all. Would you be surprised if none of the names of these authors were among the 10 most common? (Assume that authors' names are independent and follow the same probability distribution as the names of all residents.)

Independent? In 2015 , the Report on the LC Berkeley Faculty Salary Equity Study shows that 87 of the university's 222 assistant professors were women, along with 137 of the 324 associate professors and 243 of the 972 full professors. (a) What is the probability that a randomly chosen Berkeley professor (of any rank) is a woman? (b) What is the conditional probability that a randomly chosen professor is a woman, given that the person chosen is a full professor? (c) Are the rank and sex of Berkeley professors independent? How do you know?
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