Exercise 11 Name __________________
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1) Show that P(a|b Ù a) = 1
2) Calculate the following from the full joint distribution; note the bold
(random variable distribution probabilities) versus non-bold P (probability of a
value) and upper (random variable) versus lower case (value true or false) variable names:
- P(toothache) the probability that Toothache is true.
- P(Cavity) the vector of probability values for the random
variable Cavity; it has two values, list in order <true,false>.
- P(Toothache|cavity) the vector of probability values for the
random variable Toothache given that Cavity is true.
- What is the normalization constant for Question c?
- P(cavity,tootache,catch)
- P(Cavity, toothache, catch)
- P(Cavity, toothache, Øcatch)
- P(Cavity|toothache Ú catch) the
vector of probability values for the random variable Cavity given that
either Toothache or Catch is true.
- From the full joint distribution table, determine the conditional
probability tables for the following belief diagram:
Toothache
/ \
Cavity Catch
3) Three prisoners, A, B and C, are awaiting execution of one of their number
and the pardoning of the others. What is A's chances of execution given equal
likelihood of a prisoner being executed?
The guard knows who is to be executed. Prisoner A asks the guard to tell B or
C that they will be pardoned. The guard agrees and replies that B was given the
pardon message. What are A's chances of being executed now given B is freed;
state in terms of conditional probability, let Fx="x will be freed" and Ex="x
will be executed".
4) 
Expressed
as conditional probability tables,
- C - will go to college
- S - will study
- P - will party
- E - success on exams
- F - have fun
|
|
|
|
| |
|
|
| C |
P(S) |
true
false |
0.8
0.2 |
|
|
| C |
P(P) |
true
false |
0.6
0.5 |
|
| |
|
|
| S |
P |
P(E) |
true
true
false
false |
true
false
true
false |
0.6
0.9
0.1
0.2 |
|
|
| P |
P(F) |
true
false |
0.9
0.7 |
|
- Probability that will study given that go to college?
- P(E|FÙCÙØSÙØP)?
- P(CÙSÙFÙEÙP)?
- Determine whether studied or not.
5) Classify (x=1, y=2, z=3)
Training Example
| x |
y |
z |
Classification |
| 2 |
3 |
2 |
A |
| 4 |
1 |
4 |
B |
| 1 |
3 |
2 |
A |
| 2 |
4 |
3 |
A |
| 4 |
2 |
4 |
B |
| 2 |
1 |
3 |
C |
| 1 |
2 |
4 |
A |
| 2 |
3 |
3 |
B |
| 2 |
2 |
4 |
A |
| 3 |
3 |
3 |
C |
| 3 |
2 |
1 |
A |
| 1 |
2 |
1 |
B |
| 2 |
1 |
4 |
A |
| 4 |
3 |
4 |
C |
| 2 |
2 |
4 |
A |