Represent likelihood and build a uniform probability model

Learning goal: Interpret probability on the zero-to-one scale, define a sample space and calculate event probabilities when elementary outcomes are equally likely. Distinguish equal sectors or objects from categories that contain different numbers of outcomes.

Probability is a number from zero through one describing an event's likelihood under a model. In the finite models used in this lesson, zero represents an impossible event and one represents a certain event. A value near zero indicates an unlikely event; a value near one indicates a likely event. One half describes an event with the same chance of occurring as not occurring. These are likelihood statements, not promises about one trial.

A sample space lists the possible elementary outcomes. For one roll of a fair six-sided number cube labeled one through six, the sample space is {1,2,3,4,5,6}. Fair means the model assigns equal probability to the faces. Each elementary outcome has probability 1/6. A real cube's behavior can be investigated experimentally, but the mathematical calculation begins from the stated model.

An event is a set of outcomes meeting a condition. Rolling an even number includes {2,4,6}, so its probability is 3/6 = 1/2. Rolling a number greater than four includes {5,6}, giving 2/6 = 1/3. Rolling a seven is impossible in this model, while rolling a number less than seven is certain. If the labels or device change, the sample space and event probabilities may change too.

For a uniform finite model, probability equals the number of favorable elementary outcomes divided by the total number of equally likely elementary outcomes. The equal-likelihood condition matters. Counting categories without knowing their probabilities can give an incorrect result. An event with two named categories is not automatically twice as likely as one with one category unless the underlying outcomes support that comparison.

Consider an ideal spinner divided into eight equal-angle sectors, numbered one through eight. Sectors 1, 2 and 3 are blue; 4 and 5 are green; 6, 7 and 8 are orange. The model assumes each sector is equally likely. Blue has probability 3/8, green 2/8 = 1/4 and orange 3/8. The three color categories are not equally likely even though there are three names.

The full circle is the whole sample space, including every colored sector. Color does not create a separate whole for each region. Blue or green includes five of the eight sectors, giving 5/8. Not blue also includes five sectors, so its probability is 5/8. The complement of an event consists of all outcomes outside it, and event plus complement has total probability one.

Compare the spinner probabilities with one half. Green's 1/4 is below one half and is less likely than not green. Blue's 3/8 is also below one half, while blue or green's 5/8 is above one half. A single spin can still land green. An unlikely event is possible, and a likely event can fail on an individual trial.

A fair random selection from labeled objects can also support a uniform model. If five distinct names are each written once on identical slips and one is selected by a fair mixing process, each name has probability 1/5. If one name is written twice, the slips may remain equally likely but the people no longer are. Define the elementary outcomes precisely before grouping them into an event.

A model should specify how boundary cases are handled. For an ideal spinner, an exact boundary has no positive angular region; in a classroom device, decide before trials whether a pointer on a line is respun. Record the rule consistently. Do not count an ambiguous result in whichever category helps a preferred conclusion.

The sum of probabilities over the complete sample space must be one. For the color grouping, 3/8 + 2/8 + 3/8 = 8/8. If a proposed model's probabilities sum to more than one or leave out possible outcomes, review whether events overlap or the space is incomplete. This check can reveal an error before any trials are conducted.

Worked event: On the fair number cube, the event at least five includes five and six, giving 1/3. The event greater than five includes only six, giving 1/6. The words at least include the boundary; greater than excludes it, just as in inequality reasoning.

Worked category error: A student assigns one third to each spinner color because there are three colors. The sectors, not the names, are the equally likely elementary outcomes. Green occupies two of eight equal sectors while blue and orange each occupy three, so the proposed equal-color model is incorrect.

Worked complement: The spinner's probability of not orange is one minus 3/8, or 5/8. Listing sectors 1 through 5 gives the same result. Both methods preserve the complete eight-sector whole.

Model construction: Make an eight-sector spinner drawing with equal angles or use eight equally likely numbered slips with the same color mapping. List the sample space and five events, including an impossible event, a certain event and a complement. Explain why the model assigns equal probability to sectors or slips rather than automatically to color categories.

Independent performance: Design a uniform model with at least six labeled outcomes. Define three events in everyday language, identify their favorable outcomes and calculate exact probabilities. Place the results on a zero-to-one scale and interpret their relative likelihood. Include a misleading category-count solution and explain its error. A partner checks that all elementary outcomes are included once and that grouped probabilities sum appropriately.