# Short Answer Questions

1. Explain the the meaning of "prior probability," and how it differs from "probability given some evidence."
2. How can the prior probability of a hypothesis change our judgement and the signficance of some evidence? Explain and giver an example.
3. What is the base rate fallacy?

# For Further Discussion

• Ignoring the base rate. Recall the formula for determining the probability of some evidence given a hypothesis, $\Pr(H|E) = \frac{1}{1+\frac{\Pr(E|\neg H)}{\Pr(E|H)}\cdot\frac{\Pr(\neg H)}{\Pr(H)}}$
• Suppose we simply remove the base rate (or prior probabilities) from our considerations, by writing instead, $\Pr(H|E) = \frac{1}{1+\frac{\Pr(E|\neg H)}{\Pr(E|H)}}$ Explain why this is equivalent to assuming that, $\frac{\Pr(\neg H)}{\Pr(H)} = 1.$
• Show that this latter assumption implies that $$\Pr(H)=0.5$$. (Hint: Use another law of probability, which says that $$\Pr(\neg H) = 1 - \Pr(H)$$.)
• If we're considering the hypothesis $$H =$$ "Williams is a murderer," why might Williams be upset if we remove the base rate from our considerations in this way?
• Reactions to the Adams trial. In the reading on the Regina v Adams trial, you learned a number of details about the case.
• What was your own common-sense reaction to the details of this case?
• In the original trial, DNA evidence was the only evidence presented against Adams. What problems are there with using only DNA evidence against Adams, and nothing more? How can one deal with this problem?
• An appeals court granted a retrial on the basis that jurors were not appropriately informed, but rejected the use of Bayes theorem, demanding that, "Jurors evaluate evidence and reach a conclusion not on only by means of a formula, mathematical or otherwise, but by joint application of their individual common sense and knowledge of the world." Is this the correct reaction to the Adams case?
• Base rates in the Williams case. Let $$H$$ be the hypothesis that "Williams committed the murders," and let $$E$$ be the evidence, "Carpet fibers on the victim matched those in Williams' house." Suppose we accept the prosecutions data for the Williams case:
• $$\Pr(E|\neg H) = 1/8000 = 0.0001$$
• $$\Pr(E| H) = 1$$
Use a calculator and the expression of Bayes theorem above to determine the probability of $$H$$ given $$E$$, for each of the following base rates. The first line has been filled out for you already. What would you (as a juror) conclude about the likelihood of Williams' guilt on the basis of this data?
Base Rate $$\Pr(H|E)$$
$$\Pr(H)=0.5$$ $$\Pr(\neg H)=0.5$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=1$$ 0.9999
$$\Pr(H)=0.1$$ $$\Pr(\neg H)=0.9$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=9$$
$$\Pr(H)=0.01$$ $$\Pr(\neg H)=0.99$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=99$$
$$\Pr(H)=0.001$$ $$\Pr(\neg H)=0.999$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=999$$
$$\Pr(H)=0.000001$$ $$\Pr(\neg H)=0.999999$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=999999$$

• Base rates in the Adams case Do the same for the Adams case. Let $$H$$ be the hypothesis that "Adams committed the rape," and let $$E$$ be the evidence, "The Adams genetic profile matches the perpetrator's." Suppose we accept the prosecutions data for the Williams case:
• $$\Pr(E|\neg H) = 1/27,000,000 = 0.00000004$$
• $$\Pr(E| H) = 1$$
Use a calculator and the expression of Bayes theorem above to determine the probability of $$H$$ given $$E$$, for each of the following base rates. The first line has been filled out for you already. What would you conclude (as a juror) about the likelihood of Adams' guilt on the basis of this data?
Base Rate $$\Pr(H|E)$$
$$\Pr(H)=0.5$$ $$\Pr(\neg H)=0.5$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=1$$ 0.9999
$$\Pr(H)=0.1$$ $$\Pr(\neg H)=0.9$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=9$$
$$\Pr(H)=0.01$$ $$\Pr(\neg H)=0.99$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=99$$
$$\Pr(H)=0.001$$ $$\Pr(\neg H)=0.999$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=999$$
$$\Pr(H)=0.000001$$ $$\Pr(\neg H)=0.999999$$ $$\frac{\Pr(\neg H)}{\Pr(H)}=999999$$

• Seeking out the base rate fallacy. We've seen the base rate fallacy appear in the context of murder trials and pregnancy tests. Can you think of any other kinds of probabilistic reasoning in which the base rate might appear?