Assignment
Problem #1
For this problem, please use the following stochastic truth table to determine all of your algebraic translations for probabilistic claims involving {X, Y , Z}.
|
State (si)
|
X
|
Y
|
Z
|
Pr(si)
|
|
s1
|
T
|
T
|
T
|
Pr(s1) = a1
|
|
s2
|
T
|
T
|
⊥
|
Pr(s2) = a2
|
|
s3
|
T
|
⊥
|
T
|
Pr(s3) = a3
|
|
s4
|
T
|
⊥
|
⊥
|
Pr(s4) = a4
|
|
s5
|
⊥
|
T
|
T
|
Pr(s5) = a5
|
|
s6
|
⊥
|
T
|
⊥
|
Pr(s6) = a6
|
|
s7
|
⊥
|
⊥
|
T
|
Pr(s7) = a7
|
|
s8
|
⊥
|
⊥
|
⊥
|
Pr(s8) = a8
|
The goal is to prove the following general claim (the Unconditional Sure Thing Principle holds for Factor #2):
[Pr(Z | X & Y ) > Pr(Z) and Pr(Z | X & ∼Y ) > Pr(Z)] ⇒ Pr(Z | X) > Pr(Z).
That is, the goal is the prove that the following two assumptions:
(1) Pr(Z | X & Y ) > Pr(Z)
(2) Pr(Z | X & ∼Y ) > Pr(Z)
generally entail this third claim:
(3) Pr(Z | X) > Pr(Z)
In order to do this, you should follow these two steps:
Step 1. Translate claims (1)-(3) into their algebraic counterparts, using our definitions of unconditional and conditional probability (and the above table for the salient variables). That is, using:
Pr(p) ≡ ∑si|=p Pr(si) = ∑si|=p ai
Pr(p|q) =def Pr(p & q)/Pr(q), provided that Pr(q) > 0.
Step 2. Use our two general assumptions about the ai's:
(i) Each of the ai's are on [0, 1]. That is: a1, . . . , a8 ∈ [0, 1].
(ii) The ai's must sum to 1. That is: ∑8i=1 ai = 1.
to show (in algebraic terms) that whenever (1) and (2) are both true, (3) must also be true.
Problem #2
Suppose we have an urn containing 320 objects. We are going to sample a single object o at random from the urn (each individual object is equally likely to be chosen). Consider the following three statements:
- B = o is black (∼B = o is white).
- M = o is metal (∼M = o is plastic).
- S = o is a sphere (∼S = o is a cube).
Assume that these three properties are distributed according to the following probabilistic truth-table:
|
State (si)
|
B
|
M
|
S
|
Pr(wi)
|
|
s1
|
T
|
T
|
T
|
Pr(s1) = a1 = 24
320
|
|
s2
|
T
|
T
|
⊥
|
Pr(s2) = a2 = 6
320
|
|
s3
|
T
|
⊥
|
T
|
Pr(s3) = a3 = 24
320
|
|
s4
|
T
|
⊥
|
⊥
|
Pr(s4) = a4 = 42
320
|
|
s5
|
⊥
|
T
|
T
|
Pr(s5) = a5 = 33
320
|
|
s6
|
⊥
|
T
|
⊥
|
Pr(s6) = a6 = 33
320
|
|
s7
|
⊥
|
⊥
|
T
|
Pr(s7) = a7 = 47
320
|
|
s8
|
⊥
|
⊥
|
⊥
|
Pr(s8) = a8 = 111
320
|
That is, 24 of the 320 objects are black metallic spheres; 47 of the 320 objects are white plastic spheres etc. With these basic probabilities in mind, we can use our definitions of unconditional and conditional probability (on page 1) to calculate any probability in this example.
The HW is to answer the following eleven (11) questions. [Note: once you've answered questions (1)-(5), you'll have everything you need to answer questions (6)-(11). See my 03/29/16 lecture for the 3 Proposals.]
1. What is Pr(S)?
2. What is Pr(S | B)? [That is, what is Pr(S&B)/Pr(B) ?]
3. What is Pr(S | B & M)? [That is, what is Pr(S&(B&M))/Pr(B&M) ?]
4. What is Pr(B → S)? [Hint: do the truth-table for B → S to see in which of the 8 worlds B → S is true.]
5. What is Pr ((B & M) → S)? [Hint: do the truth-table for (B & M) → S to see in which worlds it is true.]
6. Is the argument ´B ∴ S‘ inductively strong, according to Proposal #1? [Hint: use your answer to (4).]
7. Is ´B ∴ S‘ inductively strong, according to Proposal #2 (Skyrms's proposal)? [Hint: use (2).]
8. Is ´B ∴ S‘ inductively strong, according to Proposal #3 (my proposal)? [Hint: use (2) and (1).]
9. Is the argument ´B & M ∴ S‘ inductively strong, according to Proposal #1? [Hint: use (5).]
10. Is ´B & M ∴ S‘ inductively strong, according to Proposal #2 (Skyrms's proposal)? [Hint: use (3).]
11. Is ´B & M ∴ S‘ inductively strong, according to Proposal #3 (my proposal)? [Hint: use (3) and (1).]