Solutions - 5040-03

(1) Define events:

Let $G$ be the event that Jim is guilty.

Let $M$ be the event that the DNA matches.

We are given that $P[M|G^c]=0.0001$.

We assume that $P[M|G]=1$.

Since there are 100,000 citizens, $P[G]=0.00001$, $P[G^c]=0.99999$.

We are asked to compute $P[G|M]$.

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(2) Set up Bayes’ Theorem:

$$\begin{array}{c}P[G|M]=P[M|G]\frac{P[G]}{P[M]}\\ \\ \gg \gg \frac{P[M|G]P[G]}{P[M|G]P[G]+P[M|G^c]P[G^c]}\end{array}$$

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(3) Plug in values:

$$\begin{array}{c}P[G|M]=\frac{P[M|G]P[G]}{P[M|G]P[G]+P[M|G^c]P[G^c]}\\ \\ \gg \gg \frac{1\cdot 0.00001}{1\cdot 0.00001+0.0001\cdot 0.99999}\gg \gg \approx 0.0909\end{array}$$