MMP.7:1 - Problem frame

Use this pattern when inference depends on how events, responses or quantities become records, and that procedure has not yet been expressed in the probability model. A feedback log may contain successes more often than failures. A timed trial can end before its event occurs. Several readings can share one calibration error. In each case, fitting a familiar distribution to the visible numbers can answer a different question from the one you intended.

Start with one possible event and follow what the observing procedure would record. Include the possibility that it leaves no record, reports an interval or shares an influence with another observation. Repeat for a contrasting event. These cases reveal what the mathematical outcome must contain before you choose its distribution.

The result is a probability law for the recorded outcome under stated assumptions, together with its relation to the quantity being inferred. It can supply a likelihood, a distribution of future records or a reason the intended inference remains ambiguous. This pattern develops probabilistic formulation within mathematical modeling. The subject practice supplies the meaning of the event, the observation procedure and plausible relations among quantities.

You need conditional probability and sums over alternatives; continuous cases also use densities and integration. A collaborator can supply those operations when you can describe the observation procedure and interpret the returned law. If an existing model already represents that procedure and answers the question, use it. When only compatible ranges are needed, C.16.IR can provide a sufficient answer without probabilities.