Question: In the context of Exercise, are the events [X = 1] and [Y = 1] independent? Justify your answer.

Exercise: The moisture content of batches of a chemical substance is measured on a scale from 1 to 3, while the impurity level is recorded as either low (1) or high (2). Let X and Y denote the moisture content and the impurity level, respectively, of a randomly selected batch. The probabilities for each of the six possible outcomes of the experiment that records X and Y for a randomly selected batch are given in the following table.

Let A and B be the events that X = 1 and Y = 1, respectively.

(a) Find the probability of A.

(b) Find the conditional probability of B given A.

(c) Find the probability mass function of the random variable X.