I've just come across this question on Isaac Physics and I have no idea where to begin. The question is as follows:
You are serving on a jury in the Crown Court. The defendant has been accused of a serious crime, however the only evidence is that their DNA is a perfect match to the perpetrator's DNA found at the crime scene.
The expert in genetic analysis tells you that the chance of a false positive (i.e. an innocent person matching that DNA) is 1 in 3000000. The prosecution lawyer says in their summing up speech that this means that as the defendant matches the DNA, the chance that they are innocent is less than 0.00004%, This means that there is a 99.99996% chance that the person is guilty, and as this is beyond reasonable doubt you and your jury colleagues should decide that the person is guilty.
Back in the jury room, the other jurors know that you have mathematical knowledge and ask you for your view on the matter.
If you think it relevant, you may also assume that the crime was definitely committed by a British person, and that the population of Britain is 67000000.
Based on the information given, what is your best estimate of the probability that the defendant is guilty? Give your answer to two significant figures.
As I said, I'm stuck in the mud with this question. I feel something to do with calculating how many innocent people in Britain would match the perpetrator's DNA would be a logical first step, although I'm not sure how I would go about doing this.
According to Bayes’s rule:
\begin{align*} &\mathbb P[\text{guilty}\,|\,\text{DNA test positive}]\\ {}={}&\frac{\mathbb P[\text{DNA test positive}\,|\,\text{guilty}]\mathbb P[\text{guilty}]}{\mathbb P[\text{DNA test positive}\,|\,\text{guilty}]\mathbb P[\text{guilty}]+\underbrace{\mathbb P[\text{DNA test positive}\,|\,\text{innocent}]}_{=1/3\mathord,000\mathord,000}\mathbb P[\text{innocent}]}. \end{align*}
Let $p_{\text G}$ denote the prior probability of guilt and $p_{\text{FN}}$ the probability of a false negative, that is: \begin{align*} p_{\text{FN}}=\mathbb P[\text{DNA test negative}\,|\,\text{guilty}]=1-\mathbb P[\text{DNA test positive}\,|\,\text{guilty}]. \end{align*} We then have \begin{align*} \mathbb P[\text{guilty}\,|\,\text{DNA test positive}]=\frac{(1-p_{\text{FN}})p_{\text{G}}}{(1-p_{\text{FN}})p_{\text{G}}+\frac{1}{3\mathord,000\mathord,000}(1-p_{\text{G}})}.\tag{$\star$} \end{align*} We know neither $p_{\text{FN}}$ nor $p_{\text{G}}$, but we can make some estimates. Note that for a fixed $p_{\text{FN}}\in(0,1)$, the fraction in ($\star$) is increasing in $p_{\text{G}}$. Since $p_{\text{G}}\geq1/67\mathord,000\mathord,000$, we can get a lower estimate for this fraction. In fact, if the DNA test is the only evidence, the defendant’s attorney may argue that the principle of presumption of innocence requires taking $p_{\text{G}}=1/67\mathord,000\mathord,000$ to be the prior probability. Plugging this into ($\star$) gives us an expression that is a function of $p_{\text{FN}}$, which takes its maximal value when $p_{\text{FN}}=0$. That maximal value is $$\frac{1\mathord,000\mathord,000}{23\mathord,333\mathord,333}\approx0\mathord.0429.$$ Therefore, the defendant is guilty with probability at most $4\mathord.29\%$.
Of course, higher values of $p_{\text{G}}$ would give much higher estimated probabilities. Accordingly, the best shot of the defendant’s counsel is to convince the jury that $p_{\text{G}}$ is low: that is, other than the incriminating DNA evidence, there is no good reason to think that the defendant is better or worse than any other British subject.