91Ó°ÊÓ

DNA evidence can be extracted from biological traces such as blood, hair, and saliva. "DNA fingerprinting" is increasingly used in the courtroom as well as in paternity testing. Given that a person is innocent, suppose that the probability of his or her DNA matching that found at the crime scene is only \(0.000001,\) one in a million. Further, given that a person is guilty, suppose that the probability of his or her DNA matching that found at the crime scene is \(0.99 .\) Jane Doe's DNA matches that found at the crime scene. a. Find the probability that Jane Doe is actually innocent, if absolutely her probability of innocence is 0.50. Interpret this probability. Show your solution by introducing notation for events, specifying probabilities that are given, and using a tree diagram to find your answer. b. Repeat part a if the unconditional probability of innocence is \(0.99 .\) Compare results. c. Explain why it is very important for a defense lawyer to explain the difference between \(\mathrm{P}\) (DNA match person innocent) and \(\mathrm{P}\) (person innocent \(\mid\) DNA match).

Step by step solution

01

Define the Events and Probabilities

Let event \( I \) represent that Jane Doe is Innocent and \( G \) represent that she is Guilty. Let event \( M \) represent that DNA matches that found at the crime scene. We know: \( P(M|I) = 0.000001 \), \( P(M|G) = 0.99 \), \( P(I) = 0.50 \) for part (a) and \( P(I) = 0.99 \) for part (b).

02

Use the Law of Total Probability

The total probability of a DNA match, \( P(M) \), is calculated using \( P(M) = P(M|I)P(I) + P(M|G)P(G) \). For part (a), plot the values: \[ P(G) = 1 - P(I) = 0.50 \] \[ P(M) = (0.000001)(0.50) + (0.99)(0.50) \]

03

Calculate Probability of Innocence Given a DNA Match

We use Bayes' Theorem: \( P(I|M) = \frac{P(M|I)P(I)}{P(M)} \). Substitute the values from Step 2 for part (a): \[ P(I|M) = \frac{(0.000001)(0.50)}{P(M)} \]

04

Repeat for Part B with Adjusted Prior Probability

Now consider \( P(I) = 0.99 \) and \( P(G) = 0.01 \) in part (b). Recalculate \( P(M) \): \[ P(M) = (0.000001)(0.99) + (0.99)(0.01) \]. Then use Bayes' Theorem: \[ P(I|M) = \frac{(0.000001)(0.99)}{P(M)} \]

05

Compare the Results

For part (a), \( P(I|M) \) is very low. In part (b), despite a high initial belief in innocence, \( P(I|M) \) is still lower than intuitively expected. This shows how prior probability impacts belief given evidence.

06

Explain the Importance

It's crucial for defense lawyers to distinguish \( P(M|I) \), the probability of a DNA match given innocence, from \( P(I|M) \), the probability of innocence given a DNA match. The latter can often be unexpectedly low, even when the former is small, due to prior probabilities.

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Key Concepts

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

Law of Total Probability

The Law of Total Probability is a fundamental concept in probability theory. It helps us find the total probability of a complex event by breaking it down into simpler, mutually exclusive events. In the context of DNA evidence, it allows us to calculate the overall probability that a DNA match could occur, accounting for the scenarios where a person could either be guilty or innocent.

• Imagine we want to know the total chance of the DNA matching the crime scene evidence.
• We consider two possibilities: Jane Doe is either innocent or guilty.

This law guides us to sum up the probabilities of the DNA match under both scenarios. For instance, if she is innocent, the probability of a DNA match is incredibly low, while it's significantly higher if she is guilty.
Overall, by weighing these individual probabilities by how likely each scenario is (innocent vs. guilty), we obtain the total probability of the DNA evidence matching, guiding our later calculations in the legal case.

Probability Theory

Probability theory is the mathematical study of chance and uncertainty. It forms the underpinning framework for calculating 'how likely' an event is to occur. It is crucial in legal settings—such as cases involving DNA evidence—where one must objectively evaluate the likelihood of scenarios.
In probability theory, probability is a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 indicates certainty. In our example, the prior probability of Jane Doe being innocent can change based on new evidence, demonstrating how probability theory adapts with additional information.

• Initially, Jane's chance of being innocent was 50%, which means the chance of being guilty was also 50%.
• Upon receiving DNA evidence, probability theory helps calculate the new probability of her innocence given this match.

By understanding this, we can better navigate decisions in courtrooms and daily life, weighing evidence and probabilities to make more informed choices.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is represented as \( P(A|B) \)—the probability of event A occurring given event B has occurred.
In our DNA scenario, conditional probability lets us determine the chance that Jane Doe is innocent after knowing her DNA matched the crime scene. This is different from simply knowing the chance of a DNA match happening irrespective of innocence or guilt.

• The match changes the context, shifting our focus to figures like \( P(I|M) \), the probability that she's innocent given her DNA matches.

This is crucial since a DNA match might seem to suggest guilt at first glance. However, understanding conditional probability shows us how such stats can be very different from initial gut reactions and why context matters so much in these legal evaluations.

DNA Evidence

DNA evidence is increasingly integral to modern legal proceedings as a powerful form of scientific proof. It involves matching DNA from a crime scene with a suspect under the assumption that DNA is unique to each individual.
However, the strength of DNA evidence can sometimes become misunderstood without considering statistical principles. For example, the probability that an innocent person's DNA might still match the sample from a crime scene could sometimes be perceived as negligible.

• In reality, misunderstanding how these probabilities work can lead to wrongful conclusions.
• That's why there is a need to properly explain concepts like \( P(M|I) \), which denote a DNA match when the person is innocent.

Understanding how DNA evidence functions within probability is crucial. It ensures that interpretations are correctly made, and accurate conclusions are drawn—thereby preventing wrongful convictions based on incorrect assumptions about the evidence.