A tetrahedral die has four faces, numbered 1-4. If the die...

A tetrahedral die has four faces, numbered $1-4$. If the die is weighted in such a way that each number is twice as likely to land facing down as the next number (1 twice as likely as 2 , 2 twice as likely as 3 , and so on) what is the probability distribution for the face landing down?

Key Concepts

Probability Mass Function

A probability mass function (PMF) assigns a probability to each discrete outcome in a sample space, ensuring that all probabilities are non-negative and sum to 1. In applications involving weighted outcomes, the PMF is derived by normalizing the weighted values to satisfy these criteria.

Geometric Progression (Ratio Relationships in Probabilities)

A geometric progression is a sequence where each term is a fixed multiple of the previous term. In the context of weighted probabilities, if each outcome’s probability is a constant multiple (e.g., twice as likely) of another's, the raw weights form a geometric sequence. This relationship helps in setting up equations that lead to the normalization step.

Weighted Probability Distribution

This concept involves assigning different probabilities to outcomes based on specified relative weights rather than assuming that each outcome is equally likely. It is key when the occurrence of one outcome is systematically more likely than another, and these weights determine the likelihood of each event in a probability model.

Normalization of Probabilities

Normalization is the process of ensuring that the sum of the probabilities of all possible outcomes equals 1. When outcomes are weighted, their unnormalized probabilities (raw weights) must be scaled by an appropriate factor so that the final probability distribution is valid.

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