Description
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 3×3 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
2
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. CptS 540 Students Only. 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.
