LPS 31 University of California Irvine Probability Exam Questions Do as the questions ask and finish the assignment for me. Thank you for your help LPS 31

LPS 31 University of California Irvine Probability Exam Questions Do as the questions ask and finish the assignment for me. Thank you for your help LPS 31 Final Exam
June 5, 2020
Instructions
The exam is worth 100 total points. Open book and notes. Feel free to use
scratch paper and/or a calculator. Please write neatly and clearly, and upload
the exam to gradescope by Monday, June 8 at 11:59pm, CA time. Best of luck!
1
Probability Axioms (10 points)
State the three Probability Axioms (don’t just name them!):
1.
2.
3.
State the Ratio Rule:
1.
2
Argument Translation (15 points)
Translate the following argument into the language of propositional logic and
use a truth table to assess its validity:
Alice either has disease X or disease Y. Also, she either has a sore throat or
else doesn’t have disease X. Alice has a sore throat. Thus, Alice has disease X.
3
Medical Arguments (25 points)
Compute the inductive probability of the following argument assuming that
epistemic probabilities defer to base rates and chances. (Hint: this is really just
a Bayes’ theorem question.)
1. One in a thousand people have disease X.
2. The chance of a positive test result is 99% for those that have disease X.
3. The chance of a positive test result is 5% for those that don’t have disease
X.
4. Alice just tested positive.
5. Thus, Alice has disease X.
4
Taking Tests (10 points)
Suppose that an exam has 10 True or False questions. Suppose that a student
who doesn’t study has a 50% chance of answering any given question correctly,
while a student who does study has an 80% chance of answering any given
question correctly. Only half of all students study for the exam. What is the
probability that Alice studied for the exam given that she answered exactly 8
questions correctly?
5
Marbles and Urns (20 points)
Suppose that a well-mixed urn contains 100 red and blue balls, in unknown
proportion. Suppose you know that there are either 80 red balls or 20 red balls.
You initially judge each of these possibilities to be equally likely and always
update your probabilities via Conditionalization when learning propositional
evidence.
1. If one ball is drawn at random and seen to be red, what is your new
probability that the urn contains 80 red balls?
2. If 5 balls are drawn randomly (with replacement) and each is seen to be
red, what is the new probability that the urn contains 80 red balls?
3. Now suppose you can tell the 5 balls drawn are all of the same color
and that you think the color is probably red but (due to the lighting)
can’t be sure. Your probability that they are all red goes to 90% while
the probability that they are all blue does to 10%. What is your new
probability that urn contains 80 red balls?
6
Measuring Accuracy (10 points)
Suppose that Alice and Bob have the following credence functions defined over
P Q P ∨Q
.9
the propositions p, q, and p∨q: Alice .8 .2
Bob .1 .3
.4
Suppose that p is true and q is false. Compute the accuracy of Alice and Bob’s
credences using the Brier score. Which of the two is more accurate?
7
Dutch Books (10 points)
Suppose that Charlie and Deborah have the following credence functions defined over the propositions p, q, and p∨q (with p and q being mutually exclusive) and that their credences in any proposition aligns with their fair price
for a gamble paying $1 on the truth of the proposition and nothing otherwise:
P Q P ∨Q
Charlie
-.1 .5
.4
Deborah 0.3 .3
0.75
1. Is Charlie Dutch bookable? If so, construct a Dutch book for him.
2. Is Deborah Dutch bookable? If so, construct a Dutch book for her.

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