Probability

Question 1

What is a proposition?

Answer 1

States a way the world is. Can be true or false.

Question 2

What are the Kolmogorov axioms?

Answer 2

Pr is a probability measure if and only if for every proposition p:

(1) Pr(p) is a real number between 0 and 1
(2) If p is certain, Pr(p)=1
(3) if p and q are incompatible then Pr(p or q)=Pr(p)+Pr(q)

Question 3

True or false: Pr(p)≥Pr(p&q)

Answer 3

True

Question 4

True or false: Pr(p)≥Pr(p or q)

Answer 4

False - Pr(p)≤Pr(p or q)

Question 5

Pr(p|q)=?

Answer 5

Pr(p|q)=Pr(p and q)/Pr(q)

Question 6

Define ‘credence’

Answer 6

Subjective probabilities or credences are degrees of belief. they manifest in betting behaviour.

Question 7

What is ‘the dutch book theorem’?

Answer 7

An agent is liable to Dutch Book (a set of bets which are guaranteed to win whatever happens) if and only if, their degrees of belief violate the probability axioms. For example if you pay 60p against £1 that it will rain and you also bet 60p that it will not rain, you are going to pay in £1.20 and win back £1 (ie a loss of 20p) whether it rains or not. Does not abide by Probabilistic coherence.

Question 8

What is the Bayes Theorem equation?

Answer 8

Pr(h|e) = (Pr(e|h) x Pr(h))/Pr(e)

where h is hypothesis and e is evidence.

Question 9

define ‘probabilistic independence’

Answer 9

Pr(p|q)=Pr(p) and Pr(q|p)=Pr(q). The probability of p has no influence on the probability of q.

Question 10

What is ‘positive dependence’?

Answer 10

Pr(p|q)>Pr(p). In such cases q makes p more likely that it would otherwise, and p makes q more likely than it would be otherwise.

Question 11

What is ‘negative dependence’?

Answer 11

Pr(p|q)

Question 12

Define ‘correlation’

Answer 12

there is a correlation between two properties F and G provided that Pr(F) and Pr(G) are probabilistically dependent.

Question 13

What is ‘simpsons paradox’?

Answer 13

If you screen using another variable, then the correlation can disappear.