# CS2710 / ISSP 2160: Homework 7

## Uncertainty and Probabilistic Reasoning (Chapters 13-14)

Assigned: November 17, 2010

Due: December 1, 2010

#### 1. Probability (15 pts)

13.8 (Russell and Norvig, p. 507)

#### 2. Probability (15 pts)

Of the entire population, 2% has a certain disease X. A test Y, which indicates whether or not a person has the disease, is not 100% accurate. If a person has the disease, there is a 6% chance that it will go undetected by the test. However, there is also a 9% chance of "false alarm" (meaning that the person does not have the disease but the test indicates otherwise). A person Z takes a test which later comes out positive (meaning that the test says he has the disease). What is the probability of this person having the disease in reality?

#### 3. Bayesian Networks (20 pts)

14.8 (Russell and Norvig, p. 561). Here is some relevant car knowledge: Icy weather is not caused by any car-related variables, but it directly affects the battery and the starter motor. The starter motor in turn directly impacts whether the car starts.

In addition, answer the following question using the original network topology (that is in Figure 14.21) rather than the extension used above.

• Give the expression for the full joint probability for: Battery=T, Radio=T, Ignition=T, Gas=F, Starts=T, Moves=F.

#### 4. More Bayesian Networks (15 pts)

Again consider the Bayesian network in Figure 14.21.

(a) Assume we want to compute the probability of the car not moving, that is P(Moves = False). Write down the expression for computing the probability from conditionals via the blind approach.

(b) Write down a more efficient expression for computing P(Moves = False) that interleaves sums and products.

(c) Can we further speed up this computation by eliminating irrelevant variables (because the summations equal 1 by definition)? Explain.

#### 5. Diagnosis using Bayesian Networks (20 pts)

Assume a Bayesian network with 5 boolean random variables, where the topology represents that Pneumonia causes Fever, Paleness, Cough, and HighWBCcount. The associated CPT are as follows:

• P(Pneumonia = True) = 0.02

• P(Fever = True|Pneumonia = True) = 0.9
• P(Fever = True|Pneumonia = False) = 0.6

• P(Paleness = True|Pneumonia = True) = 0.7
• P(Paleness = True|Pneumonia = False) = 0.5

• P(Cough = True|Pneumonia = True) = 0.9
• P(Cough = True|Pneumonia = False) = 0.1

• P(HighWBCcount = True|Pneumonia = True) = 0.8
• P(HighWBCcount = True|Pneumonia = False) = 0.5

Assume that you have the following set of symptoms: Fever and Cough are true; Paleness and HighWBCcount are false. What is the probability P(Pneumonia = T|Fever = T, Paleness = F, Cough = T,HighWBCcount = F), that is, the probability that you suffer from Pneumonia, given the symptoms? Simplify the expression as much as possible before plugging in the values.

#### 6. JavaBayes (15 pts)

Download and install JavaBayes from www.cs.cmu.edu/~javabayes. Follw the instructions to run JavaBayes.
Load an example by going to File -> Open, and then choosing an example from the Examples directory.

For this problem load the car-starts problem from the Examples/CarStarts directory. Please express answers with 4 significant digits

Part 1: Warm up (3 points)

(a) What is the probability EngineCranks cranks? (Hint: use the Query button)

(b) What is the CPT (conditional probability table) for Lights? (Hint: use the Edit Function button)

(c) What is the probability BatteryPower is Good given that the Radio is Dead? (Hint: use the Observe button to observe that the radio is dead, and then use the Query button)

Part 2: Charge (8 points)

(a) What is the probability the Alternator is OK?

(b) What is the probability the Alternator is OK given that Charge is Low?

(c) What is the probability the Alternator is OK given that there is a Leak?

(d) What is the probability the Alternator is OK given that Charge is Low and there is a Leak?

Part 3: A bit trickier (4 points)

(a). What is the probability that Radio is Dead and that Lights is NoLight, given that BatteryPower is Good?