Week 5 Tutorial: The Mathematics of the "Match"

Calculate mutual-selection probabilities, constrain the travel radius and separate peak Sunday exposure from a higher match rate.

25 March 2027

Inputs#

Bring the Platform audit and Bio experiment protocol, including the frozen A-portrait-v1 photo and supported candidate text. Preserve the exposure definitions: a Week 4 positive response is not a mutual match. Download the five-file Week 5 workbook, or use its tables in a spreadsheet or your preferred language. No app account or real profiles are needed.

Finish with The match forecast needs an audit: the interactive quiz. Six new cases include a conditional-probability diagram, a pair table and travel options. Apply the week’s model and constraint checks, then submit all six responses to unlock the score and worked explanations. The quiz is practice and carries no course marks.

Follow the 60-minute lecture selection and independent-work plan. Complete the preparation and open the workbook before Thursday. The four blocks below are the entire 90-minute tutorial; use existing lecture or study working for review and refinement. Remaining pack material belongs to the week’s independent study allocation.

90-minute tutorial#

1. 25 minutes — calculate and break the model. Use the 100-pair table to derive joint 0.06, marginal P(B) = 0.13, and reverse conditional 6/13. For twenty new opportunities, reproduce expected 1.2 and 1 − 0.94²⁰ ≈ 70.99%. Test zero opportunities and probabilities 0 and 1. Replace independence with one shared draw: preserve expectation but obtain 6% chance of any. Explain both independence assumptions.
2. 20 minutes — constrain geography. Audit the three supplied invented routes using whole-outing time and Alex’s total cost. Library is 100 minutes/$14 and feasible; Arcade is 130 minutes/$16 and fails time; Riverside is 90 minutes/$22 and fails budget. Trace the itinerary and test exactly 120 minutes/$20, then 122 minutes/$20.01. Keep an explicit reason for each failed constraint and compare the two twenty-opportunity strategies.
3. 20 minutes — compare time windows. Reproduce Sunday 8/40 and Tuesday 3/10, including count and rate ratios. Repair the misleading Sunday headline and the confounded time-experiment proposal. Declare frozen inputs, simulated allocation, complete outcome windows, fixed stop and the generator still needed. Do not treat the supplied counts as trial outcomes for that proposal.
4. 25 minutes — freeze the project target and take the quiz. Use fifteen minutes to attach evidence to your candidate’s four feature ratings. Run the unchanged 99 controls with the primary score and conservative tie rule from the benchmark. Verify illustrative ranks 2/100 at score 93.75 and 1/100 at score 100. Save the dated objective record, then exchange or self-review it for reproducibility and unsupported claims. Use the final ten minutes for the six-case quiz: about eight minutes to respond and two to review the explanations after all cases are answered.

Deliverable: Match probability model#

Commit one notebook or equivalent bundle containing the pair-table trace, repeated-model calculations, boundary checks, dependence counterexample, venue decisions and corrected Sunday claim. Include the checked invitation and a dated primary-objective record: exact candidate and photo versions, input/control fingerprints, four evidence-backed ratings, score formula, 100-entry denominator and >= tie rule. Preserve the Week 11 sensitivity plan and the baseline before revision.

Keep qualityScore and jointMatchProbability as separately labelled outputs. A literal example should retain 93.75 for the illustrative rubric vector and 0.06 for the assumed joint event; there is no conversion between them. Test invalid probabilities, missing conditioning denominators, exact time/budget boundaries and 0/0 exposure rates. Attach expected answers, actual results and a brief explanation of what a passing check cannot validate.

Worked answers are in the lecture notes and executable workbook. Correct arithmetic does not certify independence, source validity, venue reliability or the quality of your rating evidence.

Week 6 needs the same supported candidate and feasible invitation, with recipient availability still unknown and refusal terminal. Week 11 will evaluate this frozen version before making revisions.

Interactive quiz · 6 cases · about 8 minutes + 2 to review

The match forecast needs an audit

The spreadsheet has booked a table for its expected value. Check the assumptions first.

Make a decision in every case, then select Reveal answers. You can change any response before submitting. Scores and explanations stay hidden until the whole quiz is answered.

The pair counts, probabilities, routes and time windows are separate authored teaching fixtures. The final case uses the existing course benchmark, not real dating outcomes. Practice only: no course marks. Responses stay on this page; reloading starts a fresh attempt.

The cases are readable without JavaScript. Enable JavaScript to navigate, submit your responses and unlock the explanations.

Case 01 / 06The denominator changes direction

Read a conditional from the right population

Each of 200 fictional pairs supplies both selection decisions within a complete outcome window, even when one direction says no. A and B denote the two directions. You now inspect only pairs where B selected.

Complete invented pair table · counts, not probabilities

First direction	B selects	B does not
A selects	20	30
A does not	80	70

Among the pairs where B selected, what fraction also have A selecting: P(A | B)?

A20/50 = 40%, using all pairs where A selected.B20/100 = 20%, using all pairs where B selected.C20/200 = 10%, using every pair in the table.D130/200 = 65%, counting pairs with at least one selection.

Case 02 / 06One expected match requests a chair

Separate expectation from the chance of any event

Ten new opportunities each have assumed mutual-match probability 0.10. Model I makes their match indicators independent. Model S uses one shared draw: with probability 0.10 all ten match; otherwise none do. These are hypothetical models, not observations about people.

Model supplied
E[X] = sum of the individual match probabilities. For independent opportunities only, P(any) = 1 − product of (1 − p).

Which comparison is correct?

ABoth have expected count 1, so both guarantee at least one match.BBoth have expected count 1, so both have the same 65.13% chance of any match.CBoth have expected count 1; the chance of any is about 65.13% in I and 10% in S.DDependence prevents calculating an expected count for S.

Case 03 / 06The second yes has a denominator

Multiply along the conditional path

A separate model supplies P(A) = 0.40 and P(B | A) = 0.30 for one eligible fictional pair. A is selection in one direction; B is selection in the reverse direction. The second number is conditional on A, not a marginal P(B).

Model supplied
P(A and B) = P(A) × P(B | A). Multiply the result by 100 to express it as a percentage.

What is the assumed mutual-match probability as a percentage? Enter a number without the % sign.

Mutual-match probability (%)

Enter one number. A calculator and your lecture notes are welcome.

Case 04 / 06The return bus submits an invoice

Apply both whole-outing constraints

Alex has Friday 5–7 pm and $20 total. Travel times include walking, waiting and buses. Transport prices cover both legs; activity prices are Alex’s full additional cost. No extra reserve is required in this deterministic exercise. These are invented options, and the recipient’s availability is unknown.

Travel + meeting + return in minutes · transport + activity in dollars

Option	Minutes	Cost
Studio	25 + 65 + 30	$7 + $13
Loft	20 + 65 + 36	$7 + $12
Atrium	15 + 60 + 20	$7 + $13.01

Which option passes both Alex’s time and total-cost checks?

ALoft: its lower price compensates for the return time.BAtrium: its shorter outing makes the extra cent acceptable under the stated rule.CNone: finishing exactly at a limit automatically fails the stated rule.DStudio: exactly 120 minutes and $20 pass, with no spare time or money.

Case 05 / 06Sunday hires a headline writer

Compare rates with their exposure denominators

Two disjoint fictional pair pools have complete outcome windows and at most one mutual match per pair. Time was not randomly assigned. The manager divides Sunday’s match count by Tuesday’s and declares Sunday “2.25 times better”.

Separate invented window counts · not results of a time experiment

Window	Matches	Exposures
Sunday	18	120
Tuesday	8	40

Which replacement headline is supported?

ASunday has more matches but a 15% rate versus Tuesday’s 20%; these observations do not establish a time effect.BSunday’s 2.25-fold match count is also its match-rate advantage.CSunday’s threefold exposure count gives each pair three times the chance of a match.DSunday supplies 18 confirmed meetings, compared with Tuesday’s eight.

Case 06 / 06The leaderboard borrows a percent sign

Keep rubric rank separate from match probability

An illustrative candidate has ratings [4, 3, 3, 3]. In the unchanged 99-control course benchmark, one control scores higher, three tie this candidate and 95 score lower. The vector is an arithmetic exercise; actual ratings still require quoted evidence.

Model supplied
Score = 100 × sum of four ratings / 16. Rank = 1 + count of controls whose score is greater than or equal to the candidate’s. There are 100 entries including the candidate.

Which result and interpretation follow the primary benchmark?

A81.25 points and rank 2/100; discard tied controls before ranking.B81.25 points and rank 5/100; neither number is a probability of a mutual match.C81.25 points means an 81.25% chance of a mutual match.DRank 5/100 means this candidate has a 5% mutual-match probability.

Reveal answers

Answer all 6 cases to unlock your score and the worked explanations.