91Ó°ÊÓ

Forensic analysis of JFK assassination bullets. Following the assassination of President John F. Kennedy (JFK) in \(1963,\) the House Select Committee on Assassinations (HSCA) conducted an official government investigation. The HSCA concluded that although there was a probable conspiracy, involving at least one additional shooter other than Lee Harvey Oswald, the additional shooter missed all limousine occupants. A recent analysis of assassination bullet fragments, reported in the Annals of Applied Statistics (Vol. 1,2007 ), contradicted these findings, concluding that evidence used to rule out a second assassin by the HSCA is fundamentally flawed. It is well documented that at least two different bullets were the source of bullet fragments used in the assassination. Let \(E=\\{\) bullet evidence used by the HSCA\\}, \(T=\\{\) two bullets used in the assassination \(\\},\) and \(T^{C}=\) \(\\{\) more than two bullets used in the assassination \(\\}\). Given the evidence \((E),\) which is more likely to have occurred \(-\) two bullets used \((T)\) or more than two bullets used \(\left(T^{C}\right) ?\) a. The researchers demonstrated that the ratio, \(P(T \mid E) / P\left(T^{C} \mid E\right),\) is less than \(1 .\) Explain why this result supports the theory of more than two bullets used in the assassination of JFK. b. To obtain the result in part a, the researchers first showed that \(P(T \mid E) / P\left(T^{C} \mid E\right)=[P(E \mid T) \cdot P(T)] /\left[P\left(E \mid T^{C}\right) \cdot P\left(T^{C}\right)\right]\) Demonstrate this equality using Bayes's theorem.

Step by step solution

01

Understanding Bayes' Theorem

Bayes' Theorem is a fundamental principle used to calculate conditional probabilities. It is expressed as:\[ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} \]where \(P(A | B)\) is the probability of event \(A\) occurring given event \(B\) is true, \(P(B | A)\) is the probability of event \(B\) given \(A\), and \(P(A)\) and \(P(B)\) are the probabilities of events \(A\) and \(B\) occurring, respectively. The same principle applies when considering more complex scenarios like this problem.

02

Applying Bayes' Theorem to the Problem

To express \(P(T | E)\) and \(P(T^C | E)\) using Bayes' Theorem, we write:\[ P(T | E) = \frac{P(E | T) \cdot P(T)}{P(E)} \]and\[ P(T^C | E) = \frac{P(E | T^C) \cdot P(T^C)}{P(E)} \]In these expressions, \(P(E | T)\) and \(P(E | T^C)\) represent the probability of observing the evidence \(E\) under the assumption of two bullets \((T)\) and more than two bullets \((T^C)\), respectively.

03

Deriving the Probability Ratio Expression

To find \(\frac{P(T | E)}{P(T^C | E)}\), substitute the expressions from Step 2:\[ \frac{P(T | E)}{P(T^C | E)} = \frac{\frac{P(E | T) \cdot P(T)}{P(E)}}{\frac{P(E | T^C) \cdot P(T^C)}{P(E)}} \]Simplify this expression by canceling \(P(E)\) from the numerator and denominator:\[ \frac{P(T | E)}{P(T^C | E)} = \frac{P(E | T) \cdot P(T)}{P(E | T^C) \cdot P(T^C)} \]This shows how the ratio results from comparing the likelihood of the evidence under two competing scenarios.

04

Interpreting the Results

The researchers found that this ratio is less than 1, meaning:\[ \frac{P(T | E)}{P(T^C | E)} < 1 \]This inequality suggests that \(P(T^C | E) > P(T | E)\). Thus, given the evidence \(E\), it is more likely that more than two bullets were used, supporting the theory of more than two bullets being involved in the assassination.

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Key Concepts

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

Conditional Probability

Conditional probability is a foundational concept in probability theory, which allows us to determine the probability of an event given that another event has already occurred. In the context of this problem, we're interested in evaluating the likelihood of two different scenarios regarding the JFK assassination based on given evidence. To mathematically describe conditional probability, Bayes' Theorem is particularly useful. This theorem helps in updating the probability of a hypothesis, based on new evidence. It's expressed as: \[ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} \]where:

• \( P(A | B) \) is the conditional probability of event \( A \) given \( B \).
• \( P(B | A) \) is the likelihood of event \( B \) given \( A \).
• \( P(A) \) and \( P(B) \) are the probabilities of events \( A \) and \( B \) independently.

In the exercise, to assess which scenario—two bullets or more than two bullets—is more probable, we calculate \( P(T | E) \) and \( P(T^C | E) \) using conditional probability principles. This forms the basis for understanding the ratio which indicates the scenario more likely supported by the evidence at hand.

Forensic Analysis

Forensic analysis is an application of scientific methodology to solve crimes by collecting, analyzing, and interpreting physical evidence. In historical cases like the JFK assassination, forensic analysis plays a critical role. The evidence from bullet fragments, which were used by investigators, was analyzed with scientific methods to understand the shooting scenarios.
In this context, forensic experts rely on statistical methods to examine bullet fragments and trajectories, attempting to determine the number of bullets involved and the possible locations of the shooters. These analyses might involve:

• Comparing the chemical composition of bullet fragments to see if they originated from the same bullet.
• Using ballistic reconstruction to track bullet trajectories.

This specialized analysis contributes valuable insights into what likely happened during the event and is supported by mathematical techniques, such as Bayes' Theorem, to provide a probabilistic view of the evidence.

Statistical Evidence

Statistical evidence in forensic cases is critical, as it involves applying statistical methods to interpret data derived from crime scenes. For the JFK assassination, sophisticated statistical techniques were employed to evaluate bullet fragment evidence. This includes examining the likelihood of different shooting scenarios based on collected data.
Bayesian statistics come into play by helping experts understand the probability distributions of potential factors like the number of bullets. This approach quantifies uncertainty and involves calculating ratios or likelihoods—as seen in our exercise where the ratio \( \frac{P(T | E)}{P(T^C | E)} \) was calculated—to support claims. The result of this ratio being less than 1 suggested more than two bullets were involved, contrary to the initial HSCA findings.

• Statistical evidence hence bridges raw data and probabilistic models to support judicial decisions or re-evaluations of past conclusions.

JFK Assassination

The assassination of President John F. Kennedy remains one of the most controversial and investigated events in history. On November 22, 1963, President Kennedy was assassinated in Dallas, Texas, leading to numerous investigations and conspiracy theories regarding the number of shooters involved.
The official investigation by the House Select Committee on Assassinations (HSCA) initially suggested a probable conspiracy but concluded the evidence showed only one shooter, Lee Harvey Oswald. Contrary to this, newer analyses, as mentioned in the exercise, argue against this; providing evidence that more than one shooter could have been involved.
This re-examination highlights the importance of revisiting historical evidence with modern statistical tools and forensic techniques. Such historical cases also demonstrate how evolving scientific methods can yield insights that were previously overlooked, enhancing our understanding of significant events.