Montana Grade 7 Math

A probability model describes a chance process under stated assumptions. An experimental record describes what happened in a set of trials. Use both in this invented draw-and-replace example, keeping a prediction separate from an observation.

Try this original activity

A bag contains five same-size tokens: two marked A and three marked B. Assume thorough mixing and that each token is equally likely to be drawn. After each draw, return the token and mix again. An invented record of ten draws contains seven A results and three B results. Find the theoretical and experimental probabilities of A.

Work through the reasoning

The theoretical model has two favorable tokens among five equally likely tokens, so P(A) = 2/5 = 0.4, or 40%. The experimental relative frequency is 7/10 = 0.7, or 70%, from the supplied ten-draw record. The difference is 0.3, or 30 percentage points. A short random sequence can differ from its theoretical proportion; the model does not promise exactly four A results in every ten draws. Replacement keeps the bag composition unchanged before each new draw. If real records repeatedly differed across many trials, students could check mixing, token differences, and the recording method before judging the model.

Check the common mistake

Seven A results do not change the bag into seven A tokens. The theoretical denominator counts possible tokens, while the experimental denominator counts trials. The next result is not forced to be B to balance earlier results. With replacement and the stated equal-likelihood assumptions, the next draw still has probability 2/5 for A. The ten invented trials do not establish a defective process, and more trials do not guarantee an exact match at a chosen stopping point.

Try a related case

Make a paper simulation with five equally likely labeled slips, replacing each slip after recording it. Compare the relative frequency after several trial blocks with the model. Discuss variability between blocks, and keep the actual observations separate from the predicted probability.

Selected references from Montana mathematics standards

Montana’s Office of Public Instruction identifies July 1, 2026 as the implementation date for its revised mathematics standards. The selected references below were checked in the current PK–12 mathematics document linked from the OPI mathematics directory. Use the 2026 implementation document when checking a code; earlier Montana documents use a different set of references. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

OPI identifies MAST English language arts and mathematics assessment for Grades 3–8, including Grade 7. The required-assessments directory also identifies alternate assessment pathways. Follow the school’s current assigned pathway, instructional sequence, and administration directions. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration specifications or a prediction of a state performance level. Read the current OPI assessment overview.

Read OPI mathematics guidance for later guidance. Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.