Probability - 1.5 Conditional probabilities

Question 1

What is the formal definition of conditional probabilities?

Answer 1

Question 2

Show that P(.|A) is a probability measure?

Answer 2

Question 3

What is the formal definition of the law of total probability?

Answer 3

• i.e. The whole of the sample space Ω is F-measurable and is partitioned into A_{i</sup> number of sets for i ∈ I (hence the union of all sets across the index is Ω)}
• This is the sum of all joint probabilities (B n A_i) and can be read as the (chance of B occurring under scenario i) x ( the chance that scenario i is the one we’re in in the first place)

Question 4

What is the formal definiton of Bayes’ formula?

Answer 4

The denominator is P[B] and just sums the condition probabilities that B has occurred given an event A for all possible A_i –> again assuming that all A_i for i ∈ I union to the sample space.

• The numerator shows the probability that the specific event A_k has occurred and then event B occurs.

Question 5

What is a finite or countable index set?

Answer 5

Question 6

What is the proof of the law of total probability?

Answer 6

Question 7

What is the proof of the Bayes Formula?

Answer 7