To begin with, let's look at the sample space we have which consists of the outcomes 'a', 'b', 'c', 'd', 'e', and 'f'. The sample space represents all possible outcomes of an experiment. In our case, we have 6 possible outcomes, and each is equally likely.
Since the outcomes are equally likely, each event (a single outcome) has a probability of 1/6. This is because in a uniformly distributed sample space, each event shares an equal probability which is calculated by dividing 1 by the total number of outcomes. Since we have 6 outcomes, this probability is 1/6.
However, when it comes to determining the probability mass function (PMF) of a random variable X, we are met with a challenge. The PMF of a discrete random variable is the function which takes the value of the probability that the random variable is exactly equal to some value.
Without further information about how to relate the outcomes of our experiment to numerical values for the random variable X, it's impossible to create the PMF. And without the PMF, we can't determine the probabilities P(X=1.9), P(0.312), P(0<=X<12), and P(x=0 or x=12).
This is because each of these probabilities would involve knowing how the numerical values in question relate to the outcomes 'a', 'b', 'c', 'd', 'e', and 'f'. In other words, we would need to know which of these outcomes correspond to the random variable X being equal to 1.9, 0.312, or in the interval 0<=X<12, or being equal to 0 or 12.
Without this additional information, these probabilities can't be determined.
In essence, while we know the probability of any single event in our sample space is 1/6, we simply don't have enough information about the random variable X to answer the questions (a), (b), (d), and (e).