1. Suppose you are given the following full joint probability distribution over three random variables: Grade, Study, Sleep. Grade has domain {A, B, C}, Study has domain {yes, no}, and Sleep has domain {2,4,6}. Compute the probabilities below. Show your work.
Grade:
A
B
C
Study:
yes
no
yes
no
yes
no
Sleep:
6
0.10
0.05
0.08
0.06
0.03
0.08
4
0.06
0.03
0.06
0.04
0.04
0.10
2
0.02
0.01
0.04
0.02
0.06
0.12
a. P(Grade=B, Study=yes, Sleep=4).
b. P(Study=yes, Sleep=4).
c. P(Sleep=4).
d. P((Grade=C) (Sleep=6)).
e. P(Grade=A | Study=no, Sleep=2).
f. P(Grade=A | Study=yes).
2. Currently, it is estimated that the probability of someone having Coronavirus is 5%. Current tests have a false positive rate of 5% and a false negative rate of 15%. Suppose you take the Coronavirus test and the result is positive. Compute the probability a person has the virus given that they test positive. Show your work.
3. Suppose we have the 3x3 Wumpus world shown to the right. Your agent has visited locations (1,1) and (2,1). The agent observes a breeze in (2,1), but no breeze in (1,1). Given this information, we want to compute the probability of a pit in (3,1). You may use px,y and px,y as shorthand notation for Pitx,y=true and Pitx,y=false, respectively. Similarly, you may
1 use bx,y and bx,y as shorthand notation for Breezex,y=true and Breezex,y=false, respectively.
Specifically,
a. Define the sets: breeze, known, frontier and other.
b. Following the method in the textbook and lecture, compute the probability distribution P(Pit3,1 | breeze, known). Show your work.
4. Suppose you want to reduce the false positive rate of the test in question 2 so that you are at least 50% certain that a person has the Coronavirus if they test positive. What false positive rate would achieve this goal? Show your work.
