Download Problem Set and Review Questions 3 Consider the following four

Problem Set and Review Questions 3
1. Consider the following four probabilistic statements. For each statement, answer how you
would interpret the exact probabilities given below.
a) The website fivethirtyeight, which specializes in opinion polls, forecasts that Donald
Trump has an 55% probability of winning the general elections in November.
b) The Weather Channels says that there is a 20% probability of precipitation for next
Monday in Spartanburg.
c) Home pregnancy tests are 99% accurate.
d) Male condoms have a theoretical effectiveness of 98%.
2. A very sick person is rushed to the hospital. Is it more likely that this patient will die
within a week or within a year? Why?
3. If I roll two dice, what is the probability that the sum of the two numbers is going to be
7?
4. Imagine that an event has a 50% probability of happening tomorrow, and a 50%
probability of happening the day after tomorrow. Assume that these probabilities are
independent of each other. What is the probability that this event is going to happen at
least once? What is the probability that the event will happen both tomorrow and the day
after tomorrow? What is the probability that this event will not happen in the next two
days?
5. Imagine that you are working on a project that has 10 steps, and each step has a 90%
probability of being successful. The project, however, will only succeed if all ten steps
are successful. What is the probability that your project will succeed?
6. A hypothetical person has a 1% probability of having nightmares each night. What is the
probability that this person will have at least one nightmare in a period of 100 days? How
many days are necessary for this person to have a higher than 50% chance of having at
least one nightmare?
7. Suppose that 2% of all men are color-blind, and 6% of all women are color-blind. Given
this information, answer the following questions:
a) If I choose a person randomly, what is the probability that this person is color-blind?
b) If I choose a random color-blind person, what is the probability that this person is male?
c) If I choose a random color-blind person, what is the probability that this person is
female?
8. Mammograms are the most frequently used screening technology to identify breast
cancer. Suppose that out of all women who are over the age of 40 and are asymptomatic,
1% of them have breast cancer. We selected a random woman from this population to
take the mammogram. The mammogram is 90% effective. This means that, if this woman
has breast cancer, there is a 90% probability that the mammogram will correctly conclude
that she has breast cancer, and if she does not have breast cancer, there is a 90%
probability that the mammogram will correctly conclude that she does not have breast
cancer.
This woman took the mammogram and the result was positive, suggesting that she has
the disease. What is the probability that she has breast cancer?
9. Define P(A|B) as the probability of A being true given that B is true.
a) Give two examples where P(A|B) is not very different from P(B|A);
b) Give two examples (not mentioned in class) where P(A|B) is very different from P(B|A);
c) When will P(A|B) > P(A), that is, when should a piece of information B increase my
degree of belief that A is true? Hint: start with our breast cancer problem. Under what
condition (s) a positive result should cause the posterior probability to be higher than the
prior probability?
10. In our original breast cancer problem, the prior probability was equal to 1%, and the
accuracy of the mammogram was 90% for both women with and without cancer. Suppose
it is possible to develop a technology to improve the accuracy of the mammogram from
90% to 95%, but only for one group of women. In other words, we can either make the
mammogram 95% accurate for women who have breast cancer or 95% accurate for
women who do not have breast cancer. Given this information, answer the following
questions:
a) If we develop a technology that makes the mammogram 95% accurate for women with
breast cancer (the accuracy for the other group remains at 90%), what is the probability
that a woman with a positive result has breast cancer?
b) If we develop a technology that makes the mammogram 95% accurate for women who do
not have breast cancer (the accuracy for the other group remains at 90%), what is the
probability that a woman with a positive result has breast cancer?
c) If you had to choose between these two technologies, which one would you choose?
Justify your answer.
11. In conditions of uncertainty, there are two types of mistakes that we can make: a) To
believe that something (an event, statement, hypothesis, etc.) is true when it actually is
not (a false positive, or an error of commission); or b) To believe that something is not
true when it actually is (a false negative, or an error of omission).
a) Give two examples where, according to your opinion, a false positive is more costly (i.e.
worse) than a false negative;
b) Give two examples where, according to your opinion, a false negative is more costly (i.e.
worse) than a false positive;
12. If the prior probability is either 0 or 1, is there any type of evidence that can cause the
posterior probability to be different than either 0 or 1, respectively? Explain.
13. Find on the internet one real world example of either a correct or an incorrect use of
Bayesian analysis (you can also search for Bayes’ Rule, Bayes’ Theorem, base rate
fallacy, or base rate neglect). Describe the example and explain how the (correct or
incorrect) analysis was conducted.