Reasoning With Probabilities

Question 1

What are propositional language and event language?

Answer 1

Propositional language deals with things that can be true or false and is used by philosophers and logicians - What is the probability THAT a coin will land heads?

Event language deals with things that either happen or don’t and is used by statistician and probability theory texts - What is the probability OF coin landing heads?

Question 2

Define extreme probabilities, i.e. P(A)=0 and P(A)=1, in terms of necessity.

Answer 2

Zero means that proposition A is necessarily false. One means that proposition A is necessarily true.

Question 3

What is the opposite of necessary in statistical logic?

Answer 3

Contingent (i.e. Possible but can happen otherwise)

Question 4

What are mutually exclusive events?

Answer 4

When multiple events can’t occur at the same time. If any one occurs the others cannot occur.

Question 5

What is meant by the term Independence in probability theory?

Answer 5

Two events are said to be independent if the occurrence of one doesn’t influence the probability of the occurrence of the other:

• P(A)=P(A given B)
• P(B)=P(B given A)

If the contrary is true then the events are said to be dependent:

• P(A)≠P(A given B)
• P(B)≠P(B given A)

Question 6

Describe symmetrically dependency.

Answer 6

If the occurrence of A affects the probability of B, then the occurrence of B will also affect the probability of A; this doesn’t mean that the numerical values
will be the same.

This illustrates the point for a dice roll:
P(even) = 1/2, but P(even, given it’s a 2) = 1
P(2) = 1/6, but P(2, given that it’s even) = 1/3

Question 7

What is the negation rule for an event A?

Answer 7

P(not-A) = 1 minus(-) P(A)

Question 8

What is the law of the excluded middle?

Answer 8

For every proposition A, either A is true, or its negation is true, there’s no third truth value that A could have.

Question 9

What is the RESTRICTED (mutually exclusive) disjunction rule given events A and B?

Answer 9

P(A or B) = P(A) + P(B)

Question 10

What is the GENERAL (non-mutually exclusive) disjunction rule given events A and B?

Answer 10

P(A or B) = P(A) + P(B) - P(A and B)

Question 11

What is the RESTRICTED (independent) conjunction rule?

Answer 11

P(A and B) = P(A) * P(B)

Question 12

What is the GENERAL (dependent) conjunction rule?

Answer 12

P(A and B) = P(A) * P(B/A)