Chapter 9 Self-Check Quiz: Forensic DNA Statistics

Twenty-five questions — multiple choice and short answer — to check your grasp of match probabilities, likelihood ratios, mixtures, and the two fallacies. Answer before opening the key at the bottom. All numbers are illustrative teaching figures.

Multiple choice

1. A DNA "match," reported with no probability attached, is best described as: - A. Proof the defendant was at the scene - B. A statement that two profiles correspond, whose strength is unknown until a probability is added - C. A statement that the defendant is the source with 100% certainty - D. A statement about how the DNA was deposited

2. The random match probability (RMP) measures: - A. The probability the defendant is innocent - B. The probability the defendant is the source - C. The probability a random, unrelated person would share the profile by chance - D. The probability the laboratory made an error

3. Genotype frequencies at separate loci are multiplied to get a profile frequency because the loci are: - A. Always identical - B. Approximately independent - C. Always rare - D. Located on the same chromosome

4. A likelihood ratio compares the probability of the evidence under: - A. Two competing hypotheses (e.g., defendant is a contributor vs. an unknown unrelated person is) - B. The same hypothesis twice - C. Guilt versus innocence directly - D. The prosecution's and defense's closing arguments

5. For a clean single-source match with an RMP of 1 in 5 million, the likelihood ratio (defendant vs. unrelated stranger) is approximately: - A. 5 million - B. 1 in 5 million - C. 1 - D. Cannot be determined

6. An LR of exactly 1 means the evidence: - A. Strongly favors the prosecution - B. Strongly favors the defense - C. Favors neither side — it is neutral - D. Is invalid

7. The prosecutor's fallacy is the error of: - A. Computing the RMP incorrectly - B. Treating the probability of the evidence given innocence as the probability of innocence given the evidence - C. Using too many loci - D. Reporting an LR instead of an RMP

8. The defense fallacy typically: - A. Overstates the strength of a match - B. Notes that many people would match by chance and concludes the match is nearly worthless, ignoring the other evidence against this defendant - C. Confuses two conditional probabilities in the prosecution's favor - D. Refuses to report any number

9. Probabilistic genotyping software is designed primarily to: - A. Replace the crime scene investigator - B. Interpret complex or low-template DNA mixtures by computing a likelihood ratio - C. Generate DNA profiles from blood - D. Eliminate the need for a reference sample

10. PCAST's 2016 verdict on probabilistic genotyping found foundational validity: - A. For all mixtures, simple and complex - B. For no mixtures - C. For simple mixtures within validated limits, but not yet established for complex ones - D. Only for single-source samples

11. The "black box" concern about probabilistic genotyping refers to: - A. The instrument being painted black - B. Defendants being denied access to the software's source code on trade-secret grounds - C. The software running too slowly - D. The DNA being degraded

12. In the Bayesian odds form, posterior odds equal: - A. Prior odds ÷ LR - B. LR − prior odds - C. LR × prior odds - D. LR + prior odds

13. In a Bayesian analysis of DNA evidence, the prior is appropriately supplied by: - A. The forensic scientist - B. The jury, drawing on all the non-DNA evidence - C. The software - D. The judge alone

14. An LR of 1,000,000 combined with prior odds of 1 in 1,000,000 yields posterior odds of: - A. 1,000,000 to 1 - B. 1 to 1 (even odds) - C. 1 in 1,000,000 - D. 1,000,000,000 to 1

15. Which sentence may an expert properly say on the stand? - A. "There is a 1-in-8-million chance the defendant is innocent." - B. "The probability the defendant is the source is 99.99%." - C. "A random unrelated person would share this profile with a probability of about 1 in 8 million." - D. "This is a match, so he did it."

16. The single rule that keeps an expert out of both the prosecutor's fallacy and overstated individualization is: - A. Always report the smallest possible number - B. Report the strength of the evidence, never the probability of guilt - C. Never use a likelihood ratio - D. Always defer to the prosecutor

17. A "cold hit" case (suspect found only by a database search) matters statistically because it: - A. Makes the RMP smaller - B. Bears on the prior, which can be very different from a case where independent evidence pointed to the suspect first - C. Eliminates the need for a likelihood ratio - D. Proves the suspect is guilty

18. Two validated probabilistic-genotyping programs run on the same complex mixture return LRs differing by orders of magnitude. The best interpretation is that: - A. One program is broken - B. DNA evidence is worthless - C. The LR is model-dependent, not a fact of nature, especially for complex mixtures - D. The analyst committed fraud

19. A DNA match statistic speaks to whether the defendant's DNA is present; it does not speak to: - A. The reference population used - B. How or when the DNA was deposited - C. The number of loci compared - D. The allele frequencies

20. The 1968 case usually cited as the origin of the prosecutor's fallacy in court is: - A. Daubert v. Merrell Dow - B. People v. Collins - C. Frye v. United States - D. Kumho Tire v. Carmichael

Short answer

21. In one sentence each, distinguish the two conditional probabilities the prosecutor's fallacy confuses, using the four-legs/cow analogy or a comparable one.

22. A laboratory reports an LR of 2 billion "in support of the prosecution proposition." Write out both propositions ($H_p$ and $H_d$) the number must be comparing, and state one sentence the expert may not add.

23. Explain why the same DNA evidence (the same LR) can produce a near-certain conclusion in one case and a coin-flip in another. Use the words prior and posterior.

24. A complex five-person mixture at very low template is run through validated software, which returns a number. State what an honest report should say about that number and why.

25. Give one way the CSI effect can make a jury over-trust a DNA statistic and one way it can make a jury under-value one, and name the expert's safeguard in each direction.

Answer key (try the quiz first) **Multiple choice** 1. **B** — A match is a correspondence; its strength is unknown until a probability is attached (§9.1). 2. **C** — The RMP is the probability a random, unrelated person coincidentally shares the profile (§9.1). 3. **B** — Approximate independence across loci is what justifies multiplying the frequencies (§9.1). 4. **A** — An LR compares the probability of the evidence under two competing hypotheses (§9.2). 5. **A** — LR ≈ 1/RMP = 1/(1/5,000,000) = 5,000,000 for a single-source match (§9.2). 6. **C** — An LR of 1 means the evidence is equally probable under both hypotheses: neutral (§9.2). 7. **B** — Transposing the conditional: P(match | innocent) treated as P(innocent | match) (§9.4). 8. **B** — The defense fallacy ignores the other evidence that placed *this* defendant before the court (§9.4). 9. **B** — Probabilistic genotyping interprets complex/low-template mixtures and computes an LR (§9.3). 10. **C** — Foundationally valid for simple mixtures within limits; not yet established for complex ones (§9.3). 11. **B** — The "black box" is the denial of source-code access on trade-secret grounds (§9.3). 12. **C** — posterior odds = LR × prior odds (§9.5). 13. **B** — The jury supplies the prior, drawing on all non-DNA evidence; the scientist supplies the LR (§9.5). 14. **B** — 1,000,000 × (1/1,000,000) = 1, i.e., even odds (§9.5). 15. **C** — Only C reports the strength of the evidence (coincidence) rather than the probability of guilt (§9.6). 16. **B** — "Report the strength of the evidence, never the probability of guilt" (§9.6). 17. **B** — A cold hit bears on the prior, which differs from an independently chosen suspect (§9.5). 18. **C** — For complex mixtures the LR is model-dependent; disagreement reflects that, not necessarily error (§9.3). 19. **B** — The statistic addresses presence, not the mechanism or timing of deposition (§9.1, §9.6). 20. **B** — *People v. Collins* (1968) is the canonical origin of the prosecutor's fallacy in court (§9.4). **Short answer** 21. The two probabilities are P(evidence | innocent) — e.g., the RMP, the chance a random innocent person would match — and P(innocent | evidence) — the chance the defendant is innocent given the match. They differ as "the probability an animal has four legs *given it is a cow*" (≈1) differs from "the probability an animal is a cow *given it has four legs*" (low). The fallacy swaps them. (§9.4) 22. $H_p$: the defendant is a contributor to the sample. $H_d$: an unknown, unrelated person is the contributor. The expert may **not** add anything of the form "so the probability the defendant is the source/guilty is …" — that requires a prior the scientist cannot supply, and stated as a probability of guilt it is the prosecutor's fallacy. (§9.2, §9.4, §9.6) 23. The LR is only the *weight* of the evidence (the multiplier). The conclusion is the *posterior*, which equals LR × prior odds. With a weak prior (e.g., a cold hit with no other connection), even a huge LR can yield a posterior near even odds; with a strong prior (independent evidence already implicating the defendant), the same LR yields a near-certain posterior. Same evidence, different priors, different conclusions. (§9.5) 24. The report should state that the sample falls **beyond the software's validated limits** (too many contributors, too little template) and that the number is therefore unreliable and should not be presented as a valid result. "We got a number, so we'll report it" is wrong because a model run outside its validated envelope does not produce a defensible LR. (§9.3) 25. Over-trust: a jury primed by television for certainty may hear a carefully hedged probability as a verdict — the expert's safeguard is to state the limits explicitly ("this speaks to presence, not conduct; this is not a probability of guilt"). Under-value: a jury (or a defense argument) may dismiss strong evidence as mere coincidence (the defense fallacy) — the expert's safeguard is to explain the LR as strength-of-evidence and to note that the DNA combines with the other evidence, it does not stand alone. (§9.4, §9.6, and the CSI-effect theme from Ch. 1)