Consider the following frequency counts of some itemsets in a transaction database r:

freq({A}, r) = 0.405

freq({B}, r) = 0.510

freq({C}, r) = 0.303

freq({A,B}, r) = 0.380

freq({A,C}, r) = 0.256

freq({B,C}, r) = 0.197

freq({A,B,C}, r) = 0.095

Based on this information, you can use them to compute the following probabilities. (e.g. P(X) = freq(X, r).)

a)  What is the joint probability P(A and B)? What is P(A) ? P(B)?

b) What are the confidence and lift ratio of the association rules generated from the following itemsets: {A, B, C} and { ¯A, B}?

c) Compare the results of a) and b) with the frequencies freq({A}, r) and freq({B}, r). What observations can you made about the relationship of A and B in terms of independence and possible causal relationship? Please explain your answer.