Use Length Ratios for Uniform Position

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A point is uniformly chosen along a segment of length 7. A target subsegment has length 2. What is the chance of landing in the target? Candidate answers: (A) 2/5 (B) 2/7 (C) 5/7 (D) 7/2.
• Independent check inputs: A point is uniformly chosen along a segment of length 10. A target subsegment has length 3. What is the chance of landing in the target? Candidate answers: (A) 10/3 (B) 3/10 (C) 3/7 (D) 7/10.

Learn the idea

Geometry can support probability, material, density and layout models when the assumptions are explicit. Ratios and decompositions must use nonoverlapping regions, consistent dimensions and the correct uniformity or exact-fit conditions. A uniformly chosen position on a segment assigns probability proportional to subsegment length. A target’s length divided by total length then gives its chance under this continuous model.

Here is the method on the supplied teaching case. Under uniform selection, probability is target length/total length=2/7=2/7. Continuous endpoints have zero probability.

Uniform choice is essential; a biased selection method would not use the same ratio. Segment endpoints have zero probability in this continuous model, unlike individual outcomes in a finite equally likely set.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A point is uniformly chosen along a segment of length 28. A target subsegment has length 9. What is the chance of landing in the target? Candidate answers: (A) 19/28 (B) 28/9 (C) 9/19 (D) 9/28. Show the mathematical reason, not only the selected answer. Expected result: 9/28. Under uniform selection, probability is target length/total length=9/28=9/28. Continuous endpoints have zero probability.

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