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Theoretical Discrete Probability Distributions

Discrete Mathematics · Chapter 6: Random Variables and Expected ValueSection 6.3 · lesson 37 of 878 min · free with a free account

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In this lesson

Probabilities that come from an assumption, not from data.

  • Say what makes a distribution theoretical
  • Build a distribution table with a 4-step recipe
  • Model fair and deliberately weighted processes
  • Use verification to catch design errors

Transcript

Everything said in the video, section by section (1,125 words). Read along, or search it.

Random Variables and Expected Value

Theoretical Discrete Probability Distributions

This section is about theoretical distributions, whose probabilities come from an assumption about how a random process is supposed to behave. We'll see what makes a distribution theoretical, build tables with a simple four-step recipe, and see how verification catches mistakes before a model is used.

The Opening Problem

Designing a Loot Box Before Launch

Before a mobile game studio ships a new loot-box feature, its random number generator needs a precise rule for how likely each reward tier should be. That can't be a guess based on how the feature performs after launch. There's no player data yet. It has to be a decision built directly into the design. Every one of these six tiers is equally likely. Or, legendary items should occur with probability exactly zero point one. That's a theoretical, model-based distribution. The probabilities come from an assumption, not from counting what happened. Only after the feature ships would the studio compare these designed probabilities with what real players experience. That's a different question, for section six point four.

6.3.1 Model Assumptions

What Makes a Distribution Theoretical

A theoretical distribution assigns probabilities from a stated assumption about the random process. Most often it's an equally likely assumption, like a fair coin or die. But sometimes specific probabilities are assigned directly, by design, like a loot table or a weighted spinner. What makes it theoretical isn't fairness or symmetry. It's that every probability traces back to a stated assumption, reasoned about in advance, rather than a count of what actually happened. A fair die and a deliberately weighted loot table are equally theoretical in that sense. What would make either one not theoretical is replacing the assumption with observed frequencies from real trials. That's exactly the shift section six point four makes.

Table 6.3.1

A 4-Step Recipe

Building a theoretical distribution table follows the same four steps, whatever the model. First, identify the support, every possible value of X. For a fair die, that's one through six. Second, state the assumption. For the die, fair means every face is equally likely. Third, assign each probability from the assumption. Here, one sixth for each face. And fourth, verify. Every probability is at least zero, and they sum to one. Six times one sixth is one.

Worked Examples 6.3.1 and 6.3.2

Fair Coins

Start with a single toss of a fair coin. Let X be one for heads, and zero for tails. Fair means heads and tails are equally likely, so each gets one half. They sum to one, and both are non-negative. It's a valid P M F. Now three tosses, with X the number of heads. The support is zero through three. A fair coin makes all eight outcomes equally likely, one eighth each. Group them by the number of heads. Tails, tails, tails alone gives zero heads. Three outcomes give one head, three give two heads, and only heads, heads, heads gives three. So the probabilities are one eighth, three eighths, three eighths, and one eighth. They sum to eight eighths, which is one.

Worked Example 6.3.3

A Biased Coin Is Still Theoretical

Not every coin model assumes fairness. Suppose a novelty-shop trick coin is designed to land heads thirty percent of the time, by construction. That's a design assumption, decided in advance. It isn't a measurement from flipping the actual coin. So p of one, heads, is zero point three, and p of zero, tails, must be zero point seven. They sum to one, and both are non-negative. It's a valid P M F, and exactly as theoretical as the fair coin, even though the two outcomes aren't equally likely.

Worked Examples 6.3.4 and 6.3.5

A Fair Die and a Loot Table

A single fair six-sided die gives each face probability one sixth. Six times one sixth is one. Every face shares the same probability, so the distribution is uniform. Now the loot table. The studio's design document specifies six tiers, with rarer tiers deliberately less likely. The three common tiers get zero point two each. The two rare tiers get zero point one five each. And the legendary tier gets zero point one. The common tiers add to zero point six, the rare ones to zero point three, and with the legendary tier, the total is exactly one. Every value is non-negative, so it's a valid P M F. And it's every bit as theoretical as the fair die, even though the probabilities aren't equal.

Figure 6.3.1

2 Assumptions, 2 Distributions

Here are the two distributions side by side, built from different assumptions. The fair die, on the left, assumes equally likely outcomes. Every bar is the same height, one sixth. The loot-tier model, on the right, assigns unequal probabilities by design, with the legendary tier the rarest. Both sum to one. And neither is built from observed data.

Worked Example 6.3.6

Verification Scales Up

Verification isn't a formality tacked on at the end. It's the check that catches a miscalculated or mistranscribed assumption. Here's a larger model. A shipping company assumes each standard-rate package falls into one of five weight bands, with probabilities set by the rate schedule itself. The probabilities are zero point three, zero point two eight, zero point two two, zero point one two, and zero point zero eight. The first two sum to zero point five eight, the next two to zero point three four, and adding the last gives exactly one. Every value is non-negative. So it's a valid theoretical distribution. The same two-condition check scales to supports larger than a coin or a die.

Worked Example 6.3.7

Catching a Design Error

And here's verification catching a mistake. A designer drafts a table for a fair four-sided die, intending every face to be equally likely. But a transcription slip produces zero point two five, zero point two five, zero point two, and zero point two five. Those sum to zero point nine five, not one. The table fails verification, falling zero point zero five short. Since the stated assumption was a fair die, every face should carry zero point two five. Correcting p of three restores a total of one. Catching this kind of error here, before the model is used anywhere else, is exactly what verification is for.

Takeaways

Assume, Assign, Verify

To summarize. A theoretical distribution assigns probabilities from an assumption about the random process, rather than from observed data. Building one follows the same recipe for any model. Identify the support, state the assumption, assign each probability, and verify. A model doesn't have to be uniform to be theoretical. And verifying the sum catches errors before a model is used. But passing verification only means the table is internally consistent. Whether it matches reality is a separate question. Here's something to think about. The studio verifies its fair loot die perfectly. But after launch, analysts log ten thousand players, and tier six appears far less often than one sixth of the time. Does that mean the table's arithmetic was wrong? Or could a table that passes verification still fail to match what happens, for an entirely different reason? What evidence would tell the difference, and which later section supplies the tools to gather it?

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FAQ

Questions students ask us

About Theoretical Discrete Probability Distributions

What makes a probability distribution theoretical?
A distribution is theoretical when every probability comes from a stated assumption about the random process, not from counts of observed results. The assumption can be equally likely outcomes, like a fair die, or probabilities assigned by design, like a weighted coin. Replacing the assumption with observed frequencies from real trials would make it empirical instead.
How do you build a theoretical probability distribution table?
Follow four steps: identify the support (every possible value of X), state the assumption about the process, assign each probability from that assumption, and verify. For a fair die, the support is 1 through 6, each face gets \tfrac{1}{6}, and the check is that the probabilities are nonnegative and sum to 1.
Does a theoretical distribution have to be fair or uniform?
No, a theoretical distribution does not have to be uniform. A trick coin designed to land heads 30\% of the time has P(1)=0.3 and P(0)=0.7. Those probabilities are set by design, not measured from flips, so the model is just as theoretical as a fair coin, even though the outcomes are not equally likely.
How do you check whether a probability distribution table is valid?
Check that every probability is at least 0 and that all probabilities sum to exactly 1. For a fair four-sided die, a table of 0.25, 0.25, 0.2, 0.25 sums to 0.95, so it fails. A transcription slip caused it, and setting the third face to 0.25 restores the total of 1. Passing only shows the table is internally consistent, not that it matches real outcomes.

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