Unequal outcome weights

What you will learn

Build a finite sampling mechanism, identify its equally likely elementary outcomes, and sum the weights of a grouped event.

Prerequisites: Equally likely outcomes · Chips, fractions and percentages

A list of possibilities does not specify their probabilities

A sample space tells you which outcomes are possible. A probability law adds how much probability each outcome receives. Listing four labels does not establish a one-in-four chance for each. Equal probabilities require a model of the mechanism or an explicit assumption; the number of names on a menu is insufficient.

Imagine a wheel with eight equal-width slots and a uniformly selected slot. Labels 1, 2, 3 and 4 occupy one, one, two and four slots. The eight slots are equally likely elementary outcomes. The four labels group those outcomes and have probabilities 1/8, 1/8, 2/8 and 4/8. The label names are mutually exclusive and exhaustive, but they are not equally probable.

Add mass across disjoint outcomes

For the event “even label,” keep every slot labeled 2 or 4. There are one plus four such slots, so its probability is 5/8 in this particular allocation. Counting two even label names among four label names would give 1/2 and would incorrectly treat labels as uniform. Always connect the denominator to the mechanism you actually assumed.

The hands-on activity reconstructs this original allocation: 1, 1, 2 and 4 slots. Then it changes the mechanism to two slots per label while preserving the even-label event. The probability becomes 4/8. An event definition and its probability are different objects: the set of selected labels can remain fixed while its mass changes.

Coarsening is useful, but preserve what it hides

Poker hand categories are also grouped outcomes. “Pair” and “straight flush” are both category names, not equally likely physical deals. Before using favorable-count divided by total-count, establish a uniformly sampled physical space and count the grouped outcomes inside it. Changing the grouping does not make rare and common categories equally likely.

Our equal-slot wheel is a supplied idealized mechanism, not an empirical fairness claim. A real wheel could have nonuniform landing probabilities. Then slot count alone would be insufficient, and you would need the slot probabilities themselves. Do not infer a real sampling law from the appearance of the diagram or from a small observed streak.

Hands-on: Construct mass behind the labels

Allocate eight equally likely source tokens, select even labels and compare their mass before and after the mechanism changes.

Reading population workshop…

Common mistakes

• Assuming exhaustive labels must have equal probability.
• Counting label names when the sampling mechanism selects physical slots or tickets.
• Changing the sampling mechanism without updating the event probability.
• Treating an idealized uniform mechanism as verified real-world fairness.

Keep these ideas

• A sample space needs a probability law.
• Uniform elementary outcomes can form nonuniform label groups.
• Sum the mass of disjoint favorable groups.
• State why the elementary outcomes are equally likely.

Further reading

• MIT — Sample spaces and the discrete uniform law (2018) ↗

  John Tsitsiklis and Patrick Jaillet, lecture 1 annotated slides, pages 3–4, 6–7, 10 and 13. Sample spaces, complements, inclusion–exclusion and equal-element counting. Our coin/dice and toy-card activities, trials and artwork are original; no source exercises reproduced. Checked 2026-10-21.