Thinking and Deciding

128 NORMATIVE THEORY OF PROBABILITY
would themselves have to be enormous. This is why the Bayesian personalist thinks
that frequencies are relevant to probability judgment and belief formation.
This is not an unreasonable view. Take coin flips, for example. Our prior belief
about the proportion of “heads” for a coin taken out of our pocket, as we noted
earlier, is that proportions near .50 are very likely. Finding that 7 out of 10 flips
came out heads would not change our beliefs much, because our prior probability
for .50 is so much higher than our prior probability for, say, .70. If 700 out of 1,000
flips of a coin came up heads, however, we would begin to think that it was a trick
coin, as improbable as we had thought that was at the outset. If 700,000 out of
1,000,000 flips came up heads, that would practically remove all doubt. We would
be practically certain that we had a trick coin.
In sum, the personal view is able to account for our belief that relative frequency
matters. It also explains why relative frequency sometimes does not matter very
much, that is, when we have strong prior beliefs that the relative frequency is in a
small range (for example, the relative frequency of heads will be close to .50).
We can think of the personal theory as a pair of designs, one for constructing
judgments and the other for evaluating them. The main argument for the personal
theory is that it allows us to assess probabilities when we need them — that is, for
making decisions — using all of the information we have available.
More exercises on Bayes’s theorem 8
1. What is the probability of cancer if the mammogram is negative, for a case in which
p(positive|cancer) =.792, p(positive|benign) =.096, andp(cancer) =.01? (Hint:
The probability that the test is negative is 1 minus the probability that it is positive.)
2. Suppose that 1 out of every 10,000 doctors in a certain region is infected with the AIDS
virus. A test for the virus gives a positive result in 99% of those who are infected and
in 1% of those who are not infected. A randomly selected doctor in this region gets a
positive result. What is the probability that this doctor is infected?
3. In a particular at-risk population, 20% are infected with the virus. A randomly selected
member of this population gets a positive result on the same test. What is the probability
that this person is infected?
4. You are on a jury in a murder trial. After a few days of testimony, your probability for
the defendant being guilty is .80. Then, at the end of the trial, the prosecution presents a
new piece of evidence, just rushed in from the lab. The defendant’s blood type is found
to match that of blood found at the scene of the crime, which could only be the blood of
the murderer. The particular blood type occurs in 5% of the population. What should
be your revised probability for the defendant’s guilt? Would you vote to convict?
5. (Difficult) You do an experiment in which your hypothesis (H1) is that females score
higher than males on a test. You test four males and four females and you find that
all the females score higher than all the males (D). The probability of this result’s
happening by chance, if the groups did not really differ (H0), is .0016. (This is often
called the level of statistical significance.) But you want to know the probability that
8 Answers are at the end of the chapter.