Section 1.1: What is a Probability Generating Function?

Let ( X ) represent a discrete random variable, such as the outcome of a dice roll. The probability generating function (PGF) of ( X ) is defined as a power series polynomial where the coefficient of the term with index ( n ) corresponds to the probability that ( X = n ). Mathematically, this is represented as follows:

For instance, if ( X ) denotes the outcome of a fair six-sided die, where each side has an equal chance of landing face up, the PGF is established as:

Probability generating functions possess several intriguing properties.

First, evaluating the function at ( t = 0 ) yields ( P(X = 0) ):

Second, evaluating the function at ( t = 1 ) provides the sum of the probabilities of all potential outcomes, which naturally equals 1:

Third, you can derive ( P(X = k) ) from the function by differentiating it ( k ) times and then setting ( t=0 ):

A noteworthy implication of this is that if two random variables share the same PGF, they will also exhibit the same probability mass function, meaning their outcome probabilities are identical. This principle is particularly significant for casino operators.

Fourth, in line with probability laws, if you have two independent events represented by discrete random variables, the PGF for the combined outcome is the product of their respective PGFs. Thus, for independent random variables ( X ) and ( Y ):

Finally, by differentiating our PGF once and setting ( t = 1 ):

This allows us to easily calculate the expected value from the PGF.

Section 1.2: Enhancing the Game with PGFs

Now, let's revisit the PGF for a standard die, and decompose it into simpler polynomials for further exploration:

In a typical game of Craps, the total score results from rolling two independent dice. Therefore, the PGF for the sum of two standard dice can be calculated as the square of the PGF for a single die:

Next, let's explore the possibility of decomposing this function into two distinct valid PGFs whose product remains unchanged:

It's essential to confirm that these functions meet the necessary conditions:

Let’s expand our newly defined functions to investigate whether they can identify two different dice suitable for our revamped Craps game. Expanding the first function yields:

This generates a PGF for a die with faces numbered 1, 2, 2, 3, 3, and 4. For the second function, we derive:

This results in a PGF for a die with faces numbered 1, 3, 4, 5, 6, and 8. Thus, as a casino owner, you can confidently offer an alternative pair of dice labeled 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8 without affecting the game's odds.